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TREATISE 


ON 

HYDRAULICS 


BY 

MANSFIELD MERRIMAN 

* * 

Member of American Society of Civil Engineers 


Ninth Edition, Revised and Reset 

WITH THE ASSISTANCE OF 

THADDEUS MERRIMAN 

Member of American Society of Civil Engineers 


TOTAL ISSUE, FORTY THOUSAND 


O 


NEW YORK 

JOHN WILEY & SONS 

London : CHAPMAN & HALL, Limited 

* 9*3 



Copyright, 1889, 1895, 1903, 1911 

BY 


MANSFIELD MERRIMAN 


First Edition, published February, 1889 
Second Edition, published February, 1890 
Third Edition, published March, 1891 

Fourth Edition, published March, 1892; reprinted 1893 (twice), 1894 (twice) 
Fifth Edition, enlarged, published March, 1895; reprinted 1895, 1896, 1897, 1898 
Sixth Edition, published July, 1898; reprinted 1899 
Seventh Edition, published July, 1900; reprinted 1901, 1902 (twice) 
Eighth Edition, rewritten and enlarged, published May, 1903; 
reprinted 1903, 1904, 1905, 1906, 1907, 1908, 1909, 1910 

Ninth Edition, revised and reset, published December, 1911; 
reprinted, December, 1912 


All Rights Reserved 


Transfer 



June 29,1931 


«. 


« 


Composition and Electrotyping by J. S. Cushing & Co., Norwood, Mass., U. S. A. 
Printing by Scientific Press, Brooklyn, N. Y., U. S. A. 

Binding by Braunworth & Co., Brooklyn, N. Y., U. S. A. 





PREFACE TO EIGHTH EDITION 


Since the publication of the first edition of this treatise, in 
1889, many advances have been made in Hydraulics. Some of 
these have been briefly noted in later editions, but to properly 
record and correlate them it has now become necessary to 
rewrite and reset the book. In so doing the author has en¬ 
deavored to incorporate other features that have been suggested 
to him by teachers and engineers, to whom he here expresses 
his thanks. All of these suggestions could not be followed, for 
thereby the work would have been expanded to two volumes. 
Indeed the question as to what should be left out has often been 
a more difficult one than that as to what should be inserted, and 
the author has made the decision from the point of view of the 
probable benefit that may accrue to students in engineering 
colleges and to engineers in ordinary conditions of practise. 

The same plan of arrangement as in former editions has 
been followed, but two new chapters have been added, one on 
Hydraulic Instruments and Observations, which treats of the 
methods of measuring pressures and velocities, and another on 
Pumps and Pumping, in which the various machines for raising 
water are discussed from a hydraulic point of view. Among 
the new topics introduced in the other chapters may be noted 
the vortex whirl that occurs in emptying a vessel, new coeffi¬ 
cients for dams and for steel and wood pipes, the loss of head in 
pipes due to curvature, branched circuits or diversions in pipe 
systems, the influence of piers in producing backwater, canals 
for water-power plants, discharge curves for rivers, the tidal 
and the land bore, water-supply estimates, water hammer in 
pipes, the stability of a ship, and hydraulic-electric analogies. 
Vlany new examples and problems are given and in these the 
.uthor has endeavored not only to exemplify the theory of the 
ubject, but also to illustrate the conditions of actual practise. 


m 


IV 


Preface to Eighth Edition 


Historical notes and references to hydraulic literature are 
presented with greater fullness than before. . . . Many let¬ 
ters from foreign countries have urged the author to introduce 
the metric system of measures into the book. To meet this 
demand the most important data, coefficients, and formulas are 
given in both English and metric measures, the latter being 
placed at the end of each chapter; the student who follows 
these will have no occasion to transform English units, but 
may learn to think in metric units and to use them without 
hesitation. . . . The most important tables are presented both 
in the English and in the metric system, the latter not being a 
mere transformation of the former but being arranged to be 
used with metric arguments. 

In former editions of this work, as in most other books, the 
numbers of the articles, formulas, cuts, and problems were con¬ 
secutive and independent. In this edition, however, only the 
articles are numbered consecutively, while the number of any 
formula, cut, or problem agrees with that of the article, and this 
is placed at the top of the right-hand page. While the main 
purpose in rewriting the book has been to keep it abreast 
with modern progress, the attempt has also been made to pre¬ 
sent the subject more concisely and clearly than before in order 
to advance the interests of thorough education and to promote 
sound engineering practise. 










NOTE TO NINTH EDITION 


During 1903-1910 the eighth edition of this book was re¬ 
printed eight times, each impression containing some changes 
and corrections. It has now become necessary to revise and 
reset the entire book in order to more fully include the advances 
of the last decade. New matter will be found on hydraulic instru¬ 
ments, methods of measuring water, oblique weirs, submerged 
tubes, regulating devices for pipes, conduits, dams, backwater, 
rainfall, evaporation, and runoff. The tables of coefficients for 
orifices, weirs, pipes, conduits, and channels have been revised 
and extended so as to include the results of recent experi¬ 
ments. Some old matter has been -omitted or condensed, and 
a few changes in arrangement have been made. About one- 
fifth of the text is put in smaller type, so as to aid teachers 
in selecting shorter courses for their classes. The hydraulic 
tables are placed in the text in connection with the matter 
explaining them instead of being collected at the end of the 
book as before. 

In this edition all tables, figures, formulas, and problems bear 
the number of the article in which they are located, this num¬ 
ber being given in heavy type on the headline of each right- 
hand page. While the amount of matter is about six percent 
greater than that in the eighth edition, it occupies twenty pages 
less, owing to the smaller type and longer page. A subject 
index will be found at the end of the volume. The authors 
have everywhere endeavored to unify the presentation of the 
subject in a manner advantageous alike to the technical student 
and the practising engineer. 


New York, November, 1911. 
















. 















CONTENTS 


Chapter 1. Fundamental Data 

PAGES 

Art. 1. Units of Measure. 2. Physical Properties of Water. 3. The Weight 
of Water. 4. Atmospheric Pressure. 5. Compressibility of Water. G. Accel¬ 
eration due to Gravity. 7. Plistorical Notes. 8 . Numerical Computations. 

9. Data in the Metric System ......... 1-22 

Chapter 2 . Hydrostatics 

Art. 10. Transmission of Pressure. 11. Head and Pressure. 12. Loss of 
Weight in Water. 13. Depth of Flotation. 14. Stability of Flotation. 15. Nor¬ 
mal Pressure. 16. Pressure in a Given Direction. 17. Center of Pressure on 
Rectangles. 18. General Rule for Center of Pressure. 19. Pressures on Gates 
and Dams. 20. Hydrostatics in Metric Measures ..... 23-43 

Chapter 3 . Theoretical Hydraulics 

Art. 21. Laws of Falling Bodies. 22. Velocity of Flow from Orifices. 

23. Flow under Pressure. 24. Influence of Velocity of Approach. 25. The 
Path of a Jet. 26. The Energy of a Jet. 27. Impulse and Reaction of a Jet. 

28. Absolute and Relative Velocities. 29. Flow from a Revolving Vessel. 

30. Theoretic Discharge. 31. Steady Flow in Smooth Pipes. 32. Emptying 
a Vessel. 33. Computations in Metric Measures ..... 44-74 

Chapter 4 . Instruments and Observations 

Art. 34. General Considerations. 35. The Hook Gage. 36. Pressure 
Gages. 37. Differential Pressure Gages. 38. Water Meters. 39. Mean Ve¬ 
locity and Discharge. 40. The Current Meter. 41. The Pitot Tube. 42. Dis¬ 
cussion of Observations .......... 75-108 

Chapter 5 . Flow through Orifices 

Art. 43. Standard Orifices. 44. Coefficient of Contraction. 45. Coefficient 
of Velocity. 46. Coefficient of Discharge. 47. Circular Vertical Orifices. 

48 . Square Vertical Orifices. 49. Rectangular Vertical Orifices. 50. Velocity 
of Approach. 51. Submerged Orifices. 52. Suppression of the Contraction. 

53. Orifices with Rounded Edges. 54. Water Measurement by Orifices. 55. The 



Contents 


vm 


PAGES 

Miner’s Inch. 56. Loss of Energy or Head. 57. Discharge under a Drop¬ 
ping Head. 58. Emptying and Filling a Canal Lock. 59. Computations in 
Metric Measures . 109-140 

Chapter 6. Flow of Water over Weirs 

Art. 60. Standard Weirs. 61. Formulas for Discharge. 62. Velocity 
of Approach. 63. Weirs with End Contractions. 64. Weirs without End 
Contractions. 65. Francis’ Formulas. 66. Other Weir Formulas. 67. Sub¬ 
merged Weirs. 68. Rounded and Wide Crests. 69. Waste Weirs and Dams. 

70. The Surface Curve. 71. Triangular Weirs. 72. Trapezoidal Weirs. 

73. Oblique Weirs. 74. Computations in the Metric System . . . 141-176 


Chapter 7 . Flow of Water through Tubes 

Art. 75. Loss of Energy or Head. 76. Loss due to Expansion of Section. 

77. Loss due to Contraction of Section. 78. The Standard Short Tube. 

79. Conical Converging Tubes. 80. Inward Projecting Tubes. 81. Diverging 
and Compound Tubes. 82. Submerged Tubes. S3. Nozzles and Jets. 84. Lost 
Head in Long Tubes. 85. Inclined Tubes and Pipes. 86. Velocities in a 
Cross-section. 87. Fountain Flow. 88. Computations in Metric Measures 176-210 

Chapter 8. Flow of Water through Pipes 

Art. 89. Fundamental Ideas. 90. Loss of Head in Friction. 91. Loss of 
Head in Curvature. 92. Other Losses of Head. 93. Formula for Mean Ve¬ 
locity. 94. Computation of Discharge. 95. Computation of Diameter. 

96. Short Pipes. 97. Long Pipes. 98. Piezometer Measurements. 99. The 
Hydraulic Gradient. 100. A Compound Pipe. 101. A Pipe with Nozzle. 

102. House Service Pipes. 103. Operating and Regulating Devices. 104. Water 
Mains in Towns. 105. Branches and Diversions. 106. Cast Iron Pipes. 

107. Riveted Pipes. 108. Wood Pipes. 109. Fire Hose. 110. Other Formu¬ 
las for Flow in Pipes. 111. Computations in Metric Measures . . . 211-271 


Chapter 9 . Flow in Conduits 

Art. 112. Definitions. 113. Formula for Mean Velocity. 114. Circular 
Conduits, Full or Half-full. 115. Circular Conduits, partly Full. 116. Rect¬ 
angular Conduits. 117. Trapezoidal Sections. 118. Kutter’s Formula. 

119. Sewers. 120. Ditches and Canals. 121. Large Steel and Wood Pipes. 

122. Bazin’s Formula. 123. Masonry Conduits. 124. Other Formulas for 
Conduits. 125. Losses of Head. 126. Velocities in a Cross-section. 127. Com¬ 
putations in Metric Measures. 272-317 













Contents 


IX 


Chapter 10 . The Flow of Rivers 

' PAGES 

Art. 128. General Considerations. 129. Velocities in a Cross-section. 

130. Velocity Measurements. 131. Gaging the Discharge. 132. Approximate 
Gagings. 133. Comparison of Gaging Methods. 134. Variations in Discharge. 

135. Transporting Capacity of Currents. 136. Influence of Dams and Piers. 

137. Steady Non-uniform Flow. 138. The Surface Curve. 139. The Jump 
and the Bore. 140. The Backwater Curve. 141. The Drop-down Curve 318-364 

Chapter 11 . Water Supply and Water Power 

Art. 142. Rainfall. 143. Evaporation. 144. Ground Water and Runoff. 

145. Estimates for Water Supply. 146. Estimates for Water Power. 147. Water 
delivered to a Motor. 148. Effective Head on a Motor. 149. Measurement 
of Effective Power. 150. Tests of Turbine Wheels. 151. Facts concerning 
Water Power ............ 365-398 


Chapter 12 . Dynamic Pressure of Water 

Art. 152. Definitions and Principles. 153. Experiments on Impulse and 
Reaction. 154. Surfaces at Rest. 155. Immersed Bodies. 156. Curved Pipes 
and Channels. 157. Water Hammer in Pipes. 158. Moving Vanes. 159. Work 
derived from Moving Vanes. 160. Revolving Vanes. 161. Work derived from 
9 Revolving Vanes. 162. Revolving Tubes. 399-431 

Chapter 13 . Water Wheels 

Art. 163. Conditions of High Efficiency. 164. Overshot Wheels. 

165. Breast Wheels. 166. Undershot Wheels. 167. Vertical Impulse Wheels. 

168. Horizontal Impulse Wheels. 169. Downward-flow Impulse Wheels. 

170. Nozzles for Impulse Wheels. 171. Special Forms of Wheels . . 432-452 

Chapter 14 . Turbines 

Art. 172. The Reaction Wheel. 173. Classification of Turbines. 174. Re¬ 
action Turbines. 175. Flow through Reaction Turbines. 176. Theory of Re¬ 
action Turbines. 177. Design of Reaction Turbines. 178. Guides and Vanes. 

179. Downward-flow Turbines. 180. Impulse Turbines. 181. Special Devices. 

182. The Niagara Turbines. 453-484 


Chapter 15 . Naval Hydromechanics 


Art. 183. General Principles. 184. Frictional Resistances. 185. Work for 
Propulsion. 186. The Jet Propeller. 187. Paddle Wheels. 188. The Screw 
Propeller. 189. Stabil ty of a Ship. 190. Action of the Rudder. 191. Tides 
and Waves. 


485-503 




X 


Contents 


Chapter 16 . Pumps and Pumping 

PAGES 

Art. 192. General Notes and Principles. 193. Raising Water by Suction. 

194. The Force Pump. 195. Losses in the Force Pump. 196. Pumping En¬ 
gines. 197. The Centrifugal Pump. 198. The Hydraulic Ram. 199. Other 
Kinds of Pumps. 200. Pumping through Pipes. 201. Pumping through 
Hose .............. 504—538 


Appendix 

Art. 202. Hydraulic-electric Analogies. 203. Miscellaneous Problems. 

204. Answers to Problems. 205. Explanation of Tables .... 539-545 

Mathematical Tables 

Tables A and B. Fundamental Hydraulic Constants. C. Metric Equiva¬ 
lents of English Units. D. English Equivalents of Metric Units. E. Squares 
*of Numbers. F. Areas of Circles. G. Trigonometric Functions. H. Loga¬ 
rithms of Trigonometric Constants. J. Logarithms of Numbers. K. Constants 
and their Logarithms. 546-556 

Hydraulic Tables (In text) 

The number of the Table is also the number of the Article 

Table la. Inches and Feet. lb. Gallons and Cubic Feet. 3 and 9a. Weight 
of Distilled Water. 4 and 9A Atmospheric Pressure. 6 and 9c. Acceleration 
of Gravity. 11 and 20. Heads and Pressures. 22 and 33. Velocities and 
Velocity-heads. 47 a and 59#. Circular Vertical Orifices. 47A Small Circular 
Orifices. 48 and 59A Square Vertical Orifices. 49. Rectangular Vertical 
Orifices. 51. Submerged Orifices. 63 and 74#. Contracted Weirs. 64 and 
74A Suppressed Weirs. 66. Bazin’s Coefficients for Weirs. 67. Submerged 
Weirs. 68. Wide Crested Weirs. 69# and 74c. Dams. 69A Ogee Dams. 

79. Conical Tubes. 82. Submerged Tubes. 83. Vertical Jets from Nozzles. 

87. Fountain Flow from Vertical Pipes. 90# and 111#. Friction Factors for 
Pipes. 90£ and 111A Loss of Head in Pipes. 106. Friction Factors for Cast 
Iron Pipes. 114 and 127#. Circular Conduits. 115. Circular Conduits, partly 
Full. 116 and 127A Rectangular Conduits. 119 and 127c. Sewers. 120 and 
127c/. Channels in Earth. 121#. Riveted Steel Pipes. 121A Cast Iron Pipes. 

122 and 127c. Bazin’s Coefficients for Channels. 140. The Backwater Func¬ 
tion. 141. The Drop-down Function. 142. Rainfall in United States. 

143#. Evaporation from Water Surfaces. 14.3A Evaporation from Land Sur¬ 
faces. 144#. Maximum Flood Flows. 144A Observed Rainfall and Runoff. 


Index . 


. 557-565 


















TREATISE ON HYDRAULICS 


CHAPTER 1 
FUNDAMENTAL DATA 
Article 1. Units of Measure 

The unit of linear measure universally used in English and 
American hydraulic literature is the foot, which is defined as 
one-third of the standard yard. For some minor purposes, such 
as the designation of the diameters of orifices and pipes, the inch 
is employed, but inches should always be reduced to feet for use 
in hydraulic formulas. The unit of superficial measure is usually 
the square foot, except for the expression of the intensity of pres¬ 
sures, when the square inch is more commonly employed. 


Table la . Inches Reduced to Feet 


Inches 

Feet 

Inches 

Feet 

Square 

Inches 

Square 

Feet 

Cubic 

Inches 

Cubic 

Feet 

% 

0.0104 

3 

0.2500 

IO 

0.6944 

IOOO 

0.5787 

X 

.0208 

4 

•3*333 

20 

1.3889 

2000 

I-I574 

X 

•0313 

5 

.4167 

30 

2.0833 

3000 

1.7361 

X 

.0417 

6 

.5000 

40 

2.6777 

4000 

2.3148 

A 

.0521 

7 

•S 833 

50 

3.4722 

5000 

2-8935 

X 

.0625 

8 

.6667 

60 

4.1667 

6000 

3.4722 

X 

.0729 

9 

.7500 

70 

4-55oo 

7000 

4.0509 

i 

•0833 

IO* 

•8333 

80 

5-3555 

8000 

4.6296 

2 

.1667 

11 

.9167 

90 

6.2500 

9000 

5-2083 


The units of volume 'employed in measuring water are the 
cubic foot and the gallon, but the latter must always be reduced 
to cubic feet for use in hydraulic formulas. In Great Britain and 
its colonies the Imperial gallon is used, but in the United States 

1 




















2 


Chap. 1. Fundamental Data 


the old English gallon has continued to be employed, and the 
former is 20 percent larger than the latter. The following are 
the relations between the cubic foot and the two gallons: 

1 cubic foot = 6.2288 Imp. gallons = 7.481 U. S. gallons 

1 Imp. gallon = 0.1605 cubic feet = 1.201 U. S. gallons 

1 U. S. gallon = 0.1337 cubic feet = 0.832 Imp. gallons 

In this book the word “gallon” will always mean the United 
States gallon of 231 cubic inches, unless otherwise stated. 


Table 1 b . Gallons and Cubic Feet 


Cubic 

Feet 

U.S. 

Gallons 

U.S. 

Gallons 

Cubic 

Feet 

Cubic 

Feet 

Imperial 

Gallons 

Imperial 

Gallons 

Cubic 

Feet 

I 

W 

00 

1 

0.1337 

I 

6.229 

I 

0.1605 

2 

14.961 

2 

0.2674 

2 

12.458 

2 

O .321 r 

3 

22.442 

3 

0.4010 

3 

18.686 

3 

0.4816 

4 

28.922 

4 

0-5347 

4 

24915 

4 

0.6422 

5 

. 37-403 

5 

0.6684 

5 

31-144 

5 

0.8027 

6 

44.883 

6 

0.8021 

6 

37-373 

6 

0.9633 

7 

52-364 

7 

0.9358 

7 

43.602 

7 

1.1238 

8 

59.844 

8 

1.0695 

8 

49.830 

8 

1.2844 

9 

67-325 

9 

1.2031 

9 

56.059 

9 

1.4490 

10 

74.805 

10 

1.3368 

10 

62.288 

10 

1.6054 


The unit of force is the pound, or the force exerted by gravity 
at the surface of the earth on a mass of matter called the avoirdu¬ 
pois pound. This unit is also used in measuring weights and 
pressures of water. The intensity of pressure is measured in 
pounds per square foot or in pounds per square inch, as may be 
most convenient, and sometimes in atmospheres. Gages for 
recording the pressure of water are usually graduated to read 
pounds per square inch. 

The unit of time to be used in all hydraulic formulas is the 
second, although in numerical problems the time is often stated 
in minutes, hours, or days. Velocity or speed is defined as the 
space passed over by a body in one second, under the condition 
of uniform motion, so that velocities are to be always expressed 
in feet per second, or are to be reduced to these units if stated in 




























Physical Properties of Water. Art. 2 


3 


miles per hour or otherwise. Acceleration is the velocity gained 
in one second, and it is measured in feet per second per second. 

The unit of work is the foot-pound; that is, one pound lifted 
through a vertical distance of one foot. Energy is work which 
can be done; for example, a moving body has the ability to do 
a certain amount of work by virtue of its quantity of matter and 
its velocity, and this is called, kinetic energy. Again, water 
at the top of a fall has the ability to do a certain amount of work 
by virtue of its quantity and its height above the foot of the fall, 
and this is called potential energy. Potential energy changes 
into kinetic energy as the water drops, and kinetic energy is 
either changed into heat or may be transformed, by means of a 
water motor, into useful work. Power is work done, or energy 
capable of being transformed into work, in a specified time, and 
the unit for its measure is the horse-power, which is 550 foot¬ 
pounds per second. 

In French and German literature the metric system of measures 
is employed, and this is far more convenient than the English one in 
hydraulic computations. This system is understood and more or less 
used in all countries, and its universal adoption will probably occur 
during the present century, but the time has not yet come when an 
American engineering book can be prepared wholly in metric measures. 
This treatise will, therefore, mainly use the English units described 
above, but at the close of most of the chapters hydraulic data, tables, 
and empirical formulas will be given in metric measures. At the end 
of the volume will be found tables giving fundamental hydraulic 
constants and equivalents in each system of the principal units in 
the other system. 

Problem 1 . When one cubic foot of water, weighing 62I pounds, falls 
each second through a vertical height of 11 feet, what horse-power can 
be developed by a hydraulic motor which utilizes 80 percent of the energy ? 

Art. 2. Physical Properties of Water 

At ordinary temperatures pure water is a colorless liquid which 
possesses almost perfect fluidity; that is, its particles have the 
capacity of moving over each other, so that the slightest dis¬ 
turbance of equilibrium causes a flow. It is a consequence of 


4 


Chap. 1. Fundamental Data 


this property that the surface of still water is always level; also, 
if several vessels or tubes be connected, as in Fig. 2 , and water 
be poured into one of them, it rises in the others until, when 
equilibrium ensues, the free surfaces are in the same level plane. 

The free surface of water is in a different molecular condition 
from the other portions, its particles being drawn together by 

stronger attractive forces, 
so as to form what may 
be called the “skin of the 
water,” upon which insects 
may walk or a needle be 
caused to float. The skin 
is not immediately pierced by a sharp point which moves slowly 
upward toward it, but a slight elevation occurs, and this property 
enables precise determinations of the level of still water to be 
made by the hook gage (Art. 35 ). 

At about 32° Fahrenheit a great alteration in the molecular 
constitution of water occurs, and ice is formed. If a quantity 
of water be kept in a perfectly quiet condition, it is found that its 
temperature can be reduced to 20° or even to 15 0 Fahrenheit, 
before congelation takes place, but at the moment when this 
occurs the temperature rises to 32 0 . The freezing-point is hence 
not constant, but the melting-point of ice is always at the same 
temperature of 32 0 Fahrenheit or o° centigrade. 

While water freezes at 32 0 Fahrenheit, yet its maximum den¬ 
sity is reached at 39°-3 Fahrenheit. At this latter temperature 
its specific gravity is 1.0 while at 32 0 it is 0.99987. As the tem¬ 
perature rises above that of maximum density the specific gravity 
of water steadily grows smaller until the boiling-point is reached 
at 212 0 Fahrenheit when its specific gravity is 0.95865. To the 
occurrence of the maximum density at a temperature above the 
freezing-point is to be attributed the fortunate circumstance 
that ponds and streams do not freeze solid from the bottom up. 

Ice, as a rule, forms upon the surface of the water in a solid 
sheet. The rapidity with which such ice forms is dependent on 
the temperature and decreases with the thickness of the ice-sheet. 


























Physical Properties of Water. Art. 2 


t> 


The coefficient of linear expansion of ice varies from 0.0000408 
to 0.0000197 as ^e temperature varies from + 30° Fahrenheit 
to — 30° Fahrenheit.* Under certain conditions a rise in 
temperature may cause a considerable expansion, and if the 
sheet is a heavy one and expansion is prevented, the pressure 
brought to bear on any resisting surface becomes very great. 
A second variety of ice called frazil or slush ice is formed in 
rapidly flowing water when the temperature of the air is mate¬ 
rially below the freezing-point. This ice is formed in the shape 
of small needles which are carried along and deposited in quiet 
water below. Accumulations of frazil to a depth of 80 feet have 
been known.* A third variety, known as anchor ice, may of 
itself be formed directly on the bed and sides of a rapidly flow¬ 
ing stream or be increased in volume by accretions of frazil. In 
cold countries the design of hydraulic structures must take into 
account all of these three kinds of ice. 

Water is a solvent of high efficiency, and is therefore never 
found pure in nature. Descending in the form of rain, it absorbs 
dust and gaseous impurities from the atmosphere; flowing over the 
surface of the earth it absorbs organic and mineral substances. These 
affect its weight only slightly as long as it remains fresh, but when it 
has reached the sea and becomes salt, its weight is increased more than 
2 percent. The flow of water through orifices is only in a very slight 
degree affected by the impurities held in solution, but in the flow 
through pipes they often cause incrustation or corrosion which in¬ 
creases the roughness of the surface and diminishes the velocity. 

Tho capacity of water for heat, the latent heat evolved when it 
freezes, and that absorbed when it is transformed into steam need not 
be considered for the purposes of hydraulic investigations. Other 
physical properties, such as its variation in volume with the tempera¬ 
ture, its compressibility, and its capacity for transmitting pressures, 
are discussed in the following pages. The laws which govern its 
pressure, flow, and energy under various circumstances belong to the 
science of Hydraulics and form the subject-matter of this volume. 

Prob. 2 . How many degrees centigrade are equivalent to — 40° Fah¬ 
renheit ? How many degrees Fahrenheit are equivalent to — 40° centigrade 
and how many to + 40° centigrade ? 

* Barnes’s Ice Formation (New York, 1906), pp. 106; 226. 


6 


Chap. 1. Fundamental Data 


Art. 3 . The Weight of- Water 

The weight of water per unit of volume depends upon the 
temperature and upon its degree of purity. The following ap¬ 
proximate values are, however, those generally employed except 
when great precision is required : 

i cubic foot of water weighs 62.5 pounds 
1 U. S. gallon of water weighs 8.355 pounds 

These values will be used in this book, unless otherwise stated, 
in the solution of the examples and problems. 

% 

The weight per unit of volume of pure distilled water is the 
greatest at the temperature of its maximum density, 39°.3 Fah¬ 
renheit, and least at the boiling-point. For ordinary computa¬ 
tions the variation in weight due to temperature is not considered, 
but in tests of the efficiency of hydraulic motors and of pumps 
it should be regarded. The following table contains the weights 
of one cubic foot of pure water at different temperatures as de¬ 
duced by Hamilton Smith from the experiments of Rosetti.* 


Table 3. Weight of Distilled Water 


Temperature 

Fahrenheit 

Pounds per 
Cubic Foot 

Temperature 

Fahrenheit 

Pounds per 
Cubic Foot 

Temperature 

Fahrenheit 

Pounds per 
Cubic Foot 

32° 

62.42 

95° 

62.06 

160 0 

61 .OI 

35 

62.42 

100 

62 .OO 

165 

60.90 

39-3 

62.424 

105 

61.93 

170 

60.80 

45 

62.42 

no 

61.86 

175 

60.69 

50 

62.41 

115 

61.79 

180 

60.59 

55 

62.39 

120 

61.72 

185 

60.48 

60 

62.37 

125 

61.64 

190 

60.36 

65 

62.34 

130 

61.55 

195 

60.25 

70 

62.30 

135 

61.47 

200 

60.14 

75 

62.26 

140 

61.39 

205 

60.02 

80 

62.22 

I 45 

61.30 

210 

5989 

85 

62.17 

150 

61.20 

212 

59.84 

90 

62.12 

155 

61 .II 




* Hamilton Smith, Jr., Hydraulics : The Flow of Water through Orifices, over 
Weirs, and through Open Conduits and Pipes (London and New York, 1886), p. 14. 

















Atmospheric Pressure. Art. 4 


7 


Waters of rivers, springs, and lakes hold in suspension and 
solution inorganic matters which cause the weight per unit of 
volume to be slightly greater than for pure water. River waters 
are usually between 62.3 and 62.6 pounds per cubic foot, de¬ 
pending upon the amount of impurities and on the temperature, 
while the water of some mineral springs has been found to be as 
high as 62.7. It appears that, in the absence of specific informa¬ 
tion regarding a particular water, the weight 62.5 pounds per 
cubic foot is a fair approximate value to use. It also has the ad¬ 
vantage of being a convenient number in computations, for 62.5 
pounds is 1000 ounces, or is the equivalent of 62.5. 

Brackish and salt waters are always much heavier than fresh 
water. For the Gulf of Mexico the weight per cubic foot is about 
63.9, for the oceans about 64.1, while for the Dead Sea there is 
stated the value 73 pounds per cubic foot. For Great Salt Lake 
the weight of water varies from 69 to 76 pounds per cubic foot.* 
The weight of ice per cubic foot varies from 57.2 to 57.5 pounds. 
The sewage of American cities is impure water which weighs from 
62.4 to 62.7 pounds per cubic foot, but the sewage of European 
cities is somewhat heavier on account of the smaller amount of 
water that is turned into the sewers. 

Prob. 3 . How many gallons of water are contained in a pipe 3 inches 
in diameter and 12 feet long ? How many pounds of water are contained 
in a pipe 6 inches in diameter and 12 feet long ? 

Art. 4 . Atmospheric Pressure 

Torricelli in 1643 discovered that the atmospheric pressure 
would cause mercury to rise in a tube from which the air had been 
exhausted. This instrument is called the mercury barometer, 
and owing to the great density of mercury the height of the column 
required to balance the atmospheric pressure is only about 30 
inches. When water is used in the vacuum tube, the height of 
the column is about 34 feet. In both cases the weight of the 
barometric column is equal to the weight of a column of air of 
the same cross-section as that of the tube, both columns being 
measured upward from the common surface of contact. 

* Science, Oct. 21, 1910. 


8 


Chap. 1. Fundamental Data 


The atmosphere exerts its pressure with varying intensity 
as indicated by the readings of the mercury barometer. At and 
near the sea level the average reading is 30 inches, and as mercury 
weighs 0.49 pounds per cubic inch at common temperatures, the 
average atmospheric pressure is taken to be 30 X 0.49 or 14.7 
pounds per square inch. The pressure of one atmosphere is 
therefore defined to be a pressure of 14.7 pounds per square inch. 
Then a pressure of two atmospheres is 29.4 pounds per square 
inch. And conversely, a pressure of 100 pounds per square inch 
may be expressed as a pressure of 6.8 atmospheres. 

Pascal in 1646 carried a mercury barometer to the top of a 
mountain and found that the height of the mercury column de¬ 
creased as he ascended. It was thus definitely proved that the 
cause of the ascent of the liquid in the vacuum tube was due to 
the pressure of the air. Since mercury is 13.6 times heavier than 
water, a column of water should rise to a height of 30 X 13.6 = 
408 inches = 34 feet under the pressure of one atmosphere, and 
this was also found to be the case. A water barometer is imprac¬ 
ticable for use in measuring atmospheric pressures, but it is con¬ 
venient to know its approximate height corresponding to a given 
height of the mercury barometer. Table 4 shows heights of 
the mercury and water barometers, with the corresponding pres- 


Table 4 . Atmospheric Pressure 


Mercury 

Barometer 

Inches 

Pressure 
Pounds per 
Square Inch 

Pressure 

Atmospheres 

Water 

Barometer 

Feet 

Elevations 

Feet 

Boiling-point 
of Water 
Fahrenheit 

31 

15.2 

1.03 

35-1 

— 890 

2i3°.9 

30 

14.7 

1.00 

34-0 

O 

212 .2 

2Q 

14.2 

0.97 

32.9 

+ 920 

210 .4 

28 

13-7 

o -93 

3 i -7 

1880 

K> 

O 

00 

^4 

27 

13.2 

0.90 

30.6 

2870 

206.9 

26 

12.7 

0.86 

29-5 

3900 

205.0 

25 

12.2 

0.83 

28.3 

4970 

203.1 

24 

11.7 

0.80 

27.2 

6080 

201.1 

23 

II *3 

0.76 

26.1 

7240 

199.0 

22 

10.8 

0.72 

24.9 

8455 

196.9 

21 

10.3 

0.69 

23.8 

9720 * 

194.7 

20 

9.8 

0.67 

22.7 

IIOSO 

192.4 














Compressibility of Water. Art. 5 


9 


sures in pounds per square inch and in atmospheres. It also 
gives, in the fifth column, values from the vertical scale of alti¬ 
tudes used in barometric leveling which show approximate eleva¬ 
tions above sea level corresponding to barometer readings, pro¬ 
vided that the reading at sea level is 30 inches. In the last 
column are approximate boiling-points of water corresponding to 
the readings of the mercury barometer. 

The atmospheric pressure must be taken into account in many 
computations on the flow of water in tubes and pipes. It is this 
pressure that causes water to flow in syphons and to rise in tubes 
from which the air has been exhausted. By virtue of this pres¬ 
sure the suction pump is rendered possible, and all forms of in¬ 
jector pumps depend upon it to a certain degree. On a planet 
without an atmosphere many of the phenomena of hydraulics 
would be quite different from those observed on this earth. 

Prob. 4. A mercury barometer reads 30.25 inches at the foot of a hill, 
and at the same time another barometer reads 28.56 inches at the top of 
the hill. What is the difference in height between the two stations ? 


Art. 5. Compressibility of Water 


The popular opinion that water is incompressible is not justi¬ 
fied by experiments, which show in fact that it is more compress¬ 
ible than iron or even timber within the elastic limit. These 
experiments indicate that the amount of compression is directly 
proportional to the applied pressure, and that water is perfectly 
elastic, recovering its original form on the removal of the pressure. 
The decrease in the unit of volume caused by a pressure of one 
atmosphere varies, according to the experiments of Grassi, from 
0.000051 at 35 0 Fahrenheit to 0.000045 at 8o° Fahrenheit.* As 
a mean 0.00005 may be taken for this cubical unit-compression. 


A vertical column of water accordingly increases in density 
from the surface downward. If its weight at the surface be 62.5 
pounds per cubic foot, at a depth of 34 feet the weight of a cubic 

foot will be 62.5(1 +0.00005)= 62.503 pounds, 


* Grassi, Annales de chemie et physique, 1851, vol. 31, p. 437- 


10 


Chap. 1. Fundamental Data 


and at a depth of 340 feet a cubic foot will weigh 

62.5 (1 + 0.0005) = 62.53 pounds. 

The variation in weight, due to compressibility, is hence too 
small to be regarded in hydrostatic computations. 

The modulus of elasticity of volume for water is the ratio of 
the unit-stress to the cubical unit-compression, or 

E = —— = 294 000 pounds per square inch. 

0.00005 

The modulus of elasticity of volume for steel, when subjected to 
uniform hydrostatic pressure, is the same as the common modu¬ 
lus due to stress in one direction only, or E = 30 000 000 pounds 
per square inch. Hence water is about 100 times more com¬ 
pressible than steel. 

The velocity of sound or stress in any substance is given by 
the formula u = V Eg/w, where w is the weight of a cubic unit 
of the material weighed by a spring balance at the place where 
the acceleration of gravity is g (Art. 6). For water having 
w = 62.4 pounds per cubic foot at a place where g = 32.2 
feet per second per second, and E = 42 300 000 pounds per 
square foot, this formula gives u = 4670 feet per second for 
the velocity of sound, which agrees well with the results of ex¬ 
periments. 

In order to deduce the above formula for the velocity of stress it 
is necessary to use some of the fundamental principles of elementary 
mechanics and of the mechanics of elastic bodies. Let a free rigid 
body of weight W be acted upon for one second by a constant force F 
and let / be the velocity of the body at the end of one second. Let g 
be the velocity gained in one second by W when falling under the action 
of the constant force of gravity. Then, since forces are proportional 
to their accelerations, F=W. f/g , and during the second of time the 
body has moved the distance | /. Now, consider a long elastic 
bar of the length u, so that a force applied at one end will be felt at 
the other end in one second, it being propagated by virtue of the 
elasticity of the material. Let A be the area of the cross-section 
of the bar and E the modulus of elasticity of the material. When a 
constant compressive force F is applied to the bar, the shortening ul- 




Acceleration Due to Gravity. Art. 6 


11 


timately produced is 2 Fu/AE * but if this be done for one second only 
the elongation is only half this amount, since the first increment of 
stress is just reaching the other end of the bar at the end of the second. 
The center of gravity of the bar has then moved through the distance 
\ Fu/AE, and its velocity v is Fu/AE. If w is this weight of a 
cubic unit of the material, the weight W is wAu. Inserting these 
values of v and W in the above equation, there is found 


F _ Fu 
wAu A Eg 


whence 




which is the formula for the propagation of sound or stress in 
elastic materials first established by Newton. 

Prob. 5 . Compute the velocity of sound in distilled water at 35 0 and 
also at 8o° Fahrenheit. 


Art. 6. Acceleration Due to Gravity 

The motion of water in river channels, and its flow through 
orifices and pipes, is produced by the force of gravity. This force 
is proportional to the acceleration of the velocity of a body falling 
freely in a vacuum; that is, to the increase in velocity in one sec¬ 
ond. Acceleration is measured in feet per second per second, so 
that its numerical value represents the number of feet per second 
which have been gained in one second. The letter g is used to 
denote the acceleration of a falling body near the surface of the 
earth. In pure mechanics g is found in all formulas relating to 
falling bodies; for instance, if a body falls from rest through the 
height h, it attains in a vacuum a velocity equal to V 2gh. In 
hydraulics g is found in all formulas which express the laws of 
flow of water under the influence of gravity. 

The quantity 32.2 feet per second per second is an approxi¬ 
mate value of g which is often used in hydraulic formulas. It is, 
however, well known that the force of gravity is not of constant 
intensity over the earth’s surface, but is greater at the poles than 
at the equator, and also greater at the sea level than on high 
mountains. The following formula of Peirce, which is partly 
theoretical and partly empirical, gives g in feet per second per 

* Merriman’s Mechanics of Material (New York, 1911), pp. 25, 325. 





12 Chap. 1. Fundamental Data 

second for any latitude /, and any elevation e above the sea level, 
e being in feet: 

g = 32.0894(1 +0.0052375 sin 2 /) (1 — 0.000000095 7 e) (6)! 
and from this its value may be computed for any locality. 

The greatest value of g is at the sea level at the pole, and 
for this locality l = go°, e = o, whence g = 32.258. The least 
value of g is on high mountains at the equator; for this there 
may be taken l = o°, e = 10 000 feet, whence g = 32.059. The 
mean of these is the value of the acceleration used in this book, 
unless otherwise stated, namely, 

g = 32.16 feet per second per second, 

and from this the mean values of the frequently occurring 
quantities V2 g and 1/2 g are found to be 

V2g= 8.020, l/2g = O.OIS55. (6)2 

If greater precision be required, which will sometimes be the case, 
g can be computed from the above formula for the particular 
latitude and elevation. Table 6 gives multiples of the quantities 
g, 2g, 1/2 g, and V2g which will often be useful in numerical 
computations. 

Table 6. Acceleration of Gravity 


No. 

Multiples 
of g 

Multiples 
of 2g 

Multiples 
of l/2g 

Multiples 
of %/ 2 g 

No 

1 

32.16 

64.32 

0-01555 

8.02 

1 

2 

64.32 

128.6 

0.03109 

16.04 

2 

3 

96.48 

193.O 

0.04664 

24.06 

3 

4 

128.6 

257-3 

0.06219 

32.08 

4 

5 

160.8 

321.6 

0.07774 

40.10 

5 

6 

193.O 

385-9 

0.09328 

48.12 

6 

7 

225.1 

450.2 

0.1088 

56.14 

7 

8 

257-3 

5H-5 

O.1244 

64.16 

8 

9 

289.4 

578.9 

O.1399 

72.l8 

9 

10 

321.6 

643.2 

0.1555 

80.20 

10 


Prob. 6. Compute to four significant figures the values of g and 
V2g for the latitude of 4o°36 f and the elevation 400 feet. Also for 
the same latitude and the elevation 4000 feet. 

















Historical Notes. Art. 7 


13 


Art. 7. Historical Notes 

Hydraulics is that branch of the mechanics of fluids which 
treats of water in motion, while Hydrostatics treats of water at 
rest. These two branches are sometimes regarded as a part of 
Hydromechanics, the name of the mechanics of fluids and gases. 
While the main purpose of this book is to treat of water in motion, 
the most important principles of hydrostatics will also be discussed, 
since these are .necessary for a complete development of the laws 
of flow. The word “Hydraulics” is hence here used as closely 
synonymous with the hydromechanics of water. 

Hydraulics is a modern science which is still far from perfect. 
Archimedes, about 250 b.c., established a few of the principles 
of hydrostatics and showed that the weight of an immersed body 
is less than its weight in air by the weight of the water that it 
displaces. Chain and bucket pumps were used at this period by 
the Egyptians, and the force pump was invented by Ctesibius 
about 120 b.c. The Romans built aqueducts as early as 300 b.c., 
and later used earthen and lead pipes to convey water from them 
to their houses. They knew that water would rise in a lead pipe 
to the same level as in the aqueduct and that a slope was neces¬ 
sary to cause flow in the latter, but had no conception of such a 
simple quantity as a cubic foot per minute. Even this slight 
knowledge was lost after the destruction of Rome, 475 a.d., and 
Europe, for a thousand years sunk in barbarism, made no scien¬ 
tific inquiries until the Renaissance period began. 

Galileo, in 1630, studied the subject of the flotation of bodies 
in water, and a little later his pupils Castelli and Torricelli made 
notable discoveries, the former on the flow of water in rivers 
and the latter on the height of a jet issuing from an orifice. 
Pascal, about 1650, extended Torricelli’s researches on the 
influence of atmospheric pressure in causing liquids to rise in 
a vacuum. Mariotte, about 1680, considered the influence of 
friction in retarding the flow in pipes and channels, and New¬ 
ton, in 1685, observed the contraction of a jet issuing from an 
orifice. 


14 


Chap. 1. Fundamental Data 


During the eighteenth century notable advances were made. 
Daniel and John Bernoulli extended the theory of the equilibrium 
and motion of fluids, and this theory was much improved and 
generalized by D’Alembert. Bossut and Dubuat made experi¬ 
ments on the flow of water in pipes and deduced practical coeffi¬ 
cients, while Chezy and Prony, near the close of the century, 
established general formulas for computing velocity and discharge. 

During the nineteenth century progress in every branch of 
hydraulics was great and rapid. Eytelwein, Weisbach, and 
Hagen stood high among German experimenters; Venturi and 
Bidone among those of Italy; Poncelet, Darcy, and Bazin among 
those of France; while Kutter in Switzerland, Rankine in Eng¬ 
land, and James B. Francis and Hamilton Smith in America also 
took high rank for either practical or theoretical investigations. 
By the experiments and discussions of these and many other en¬ 
gineers the necessary coefficients for the discussion of orifices, 
weirs, jets, pipes, conduits, and rivers have been determined and 
the theory of the flow of water has been much extended and per¬ 
fected. The invention of the turbine by Fourneyron in 1827 
exerted much influence upon the development of water power, 
while the studies necessary for the construction of canals and for 
the improvement of rivers and harbors have greatly promoted 
hydraulic science. In this advance the engineers of the United 
States did much good work during the latter part of the nineteenth 
and are continuing it during the present part of the twentieth 
century, as is shown by the numerous valuable papers published 
in the Transactions of the American engineering societies and in 
the scientific press, many of which will be cited in this book. 

Galileo said in 1630 that the laws controlling the motion of 
the planets in their celestial orbits were better understood than 
those governing the motion of water on the surface of the earth. 
This is true today, for the theory of the flow of water in pipes 
and channels has not yet been perfected. Experiment is now 
in advance of theory, but it is intended to present both in this 
volume as far as practicable, for each is necessary to a satisfac¬ 
tory understanding of the other. 


Numerical Computations. Art. 8 15 

Prob. 7 . Who was the author of a book called Lowell Hydraulic Ex¬ 
periments? When and where was it published? What influence has it 
exerted upon hydraulic science? 

Art. 8. Numerical Computations 

The numerical work of computation should not be carried 
to a greater degree of refinement than the data of the problem 
warrant. For instance, in questions relating to pressures, the 
data are uncertain in the third significant figure, and hence more 
figures than three in the final result must be delusive. Thus 
let it be required to compute the number of pounds of water in a 
box containing 307.37 cubic feet. Taking the mean value 62.5 
pounds as the weight of one cubic foot, the multiplication gives 
the result 19 210.625 pounds, but evidently the decimals here have 
no precision, since the last figure in 62.5 is not accurate, and is 
likely to be less than 5, depending upon the impurity of the water 
and its temperature. The proper answer to this problem is 
19 200 pounds, or perhaps 19 210 pounds, and this is to be re¬ 
garded as a probable average result rather than an exact quantity. 

Three significant figures are usually sufficient in the answer 
to any hydraulic problem, but in order that the last one may be 
correct four significant figures should be used in the computa¬ 
tions. Thus, 307.37 has five significant figures and this should be 
written 307.4 before multiplying it by 62.5. The zeros following 
a decimal point of a decimal are not counted significant figures; 
thus, 0.0019 h as two an d 0.0003742 has four significant figures. 

The use of logarithms is to be recommended in hydraulic 
computations, as thereby both mental labor and time are saved. 
Four-figure tables are sufficient for common problems, and their 
use is particularly advantageous in all cases where the data are 
not precise, as thus the number of significant figures in final 
results is kept at about three, and hence statements implying 
great precision, when none really exists, are prevented. The 
four-place logarithmic table at the end of this volume will be found 
very convenient in solving numerical problems. As an example, 
let it be required to find the weight of a column of water 2.66 


16 


Chap. 1. Fundamental Data 


inches square and 28.7 feet long. The computation, both by 
common arithmetic and by logarithms, is as follows, and it will 
be found, by trying similar problems, that in general the use of 

By Arithmetic By Logarithms 


2.66 

0.04914 

2.66 0.4249 

2.66 

28.7 

2 

5-32 

9828 

0.8498 

1 596 

39312 

144 2.1584 

i6o|144 

3439 

2.6914 

7.076(0.04914 

1.410 

28.7 1.4579 

576 

62.5 

62.5 1.7959 

1316 

846 

Ans. 88.1 1.9452 

1296 

282 


20 

70 


14 Ans. 

88.1 pounds. 



6 

logarithms effects a saving of time and labor. The common 
slide rule, which is constructed on the logarithmic principle, will 
also be found very useful in the numerical work of many 
hydraulic problems. 

The tables of constants, squares, and areas of circles at the 
end of this volume will also be advantageous in abridging com¬ 
putations. For instance, it is seen at once from Table E that the 
square of 2.66 to four significant figures is 7.076, while Table F 
shows that the area of a circle having a diameter of 0.543 inch is 
0.2316 square inch. Logarithms of hydraulic and mathematical 
constants are given in Tables A, C, and K. Tables 1 a, 1 b, and 
6 of this chapter and others in the next chapter give multiples of 
constants which may be advantageously used when it is necessary 
to multiply several numbers by the same constant. For example, 
when it is required to reduce 333.4, 318.7, and 98.6 cubic feet 
to U. S. gallons, the book is opened at Table lb , where the multi¬ 
ples of 7.481 are given, and the work is as follows: 


333-4 

318.7 

98.6 

2244.2 

2244.2 

673.2 

224.4 

74.8 

59-8 

22.4 

59-8 

4-5 

3 -o 

5-2 

737-5 

2494.0 

2384.0 






















Numerical Computations. Art. 8 


17 


These results are more accurate than can be obtained with four- 
place logarithmic tables. The logarithmic work for this case 
would be the following : 


333-4 

3 * 8.7 

98.6 

2.5229 

2.5034 

1-9939 

0.8740 

0.8740 

0.8740 

33969 

3-3774 

2.8679 

2494 

2384 

737 


As this book is mainly intended for the use of students in 
technical schools, a word of advice directed especially to them 
may not be inappropriate. It will be necessary for students, in 
order to gain a clear understanding of hydraulic science, or of 
any other engineering subject, to solve many numerical problems, 
and in this a neat and systematic method should be cultivated. 
The practice of performing computations on any loose scraps of 
paper that may happen to be at hand should be at once discon¬ 
tinued by every student who has followed it, and he should here¬ 
after solve his problems in a special book provided for that pur¬ 
pose, and accompany them by such explanatory remarks as may 
seem necessary in order to render the solutions clear. Such a 
note-book, written in ink, and containing the fully worked out 
solutions of the examples and problems given in these pages, 
will prove of great value to every student who makes it. Before 
beginning the solution of a problem a diagram should be drawn 
whenever it is possible, for a diagram helps the student to clearly 
understand the problem, and a problem thoroughly understood 
is half solved. Before commencing the numerical work, it is also 
well to make a mental estimate of the final result. 

In this volume Greek letters are used only for signs of operation 


and for angles. 

The letter 3 is employed as the 

symbol of differenti- 

ation and it should be called 

“ differential.” 

Following are names 

of some Greek letters: 



a Alpha 

y) Eta 

v Nu 

Phi 

/3 Beta 

6 Theta 

7 T Pi 

if/ Psi 

y Gamma 

k Kappa 

p Rho 

£ Zeta 

3 Delta 

\ Lambda 

0- Sigma 

w Omega 

e Epsilon 

fj. Mu 

r Tau 









18 


Chap. 1. Fundamental Data 


In every rational algebraic equation it is necessary that all the 
terms should be of the same dimension, for it is impossible to add 
together quantities of different kinds. This principle will be of great 
assistance to the student in checking the correctness of algebraic work. 
For example, let a and b represent areas and l a length; then such 
an equation as al — l 2 = b is impossible, because al is a volume, while 
1 2 and b are areas. Again, let V represent velocity, Q cubic feet per 
second, and a area; then the equation Q = aV is correct dimensionally, 
for the dimension of V is length per second and hence a V is of the 
same dimension as Q. The equation Q/a= V 2 is, however, impossible, 
for Q/a is of the same dimension as the first power of V, and this can¬ 
not also be equal to its second power. 

Prob. 8. When the height of the water barometer is 33.5 feet, what 
is the height of the mercury barometer, and what is the atmospheric 
pressure in pounds per square inch? 

Art. 9 . Data in the Metric System 

When the metric system is used for hydraulic computations, 
the meter is taken as the unit of length, the cubic meter as the 
unit of volume, and the kilogram as the unit of force and weight. 
Lengths are sometimes expressed in centimeters and volumes in 
liters, but these should be reduced to meters and cubic meters 
for use in the formulas. The unit of time is the second, the unit 
of velocity is one meter per second, and accelerations are measured 
in meters per second per second. Pressures are usually expressed 
in kilograms per square centimeter and densities in kilograms per 
cubic meter. The metric horse-power is 75 kilogram-meters 
of work per second, and this is about 1} per cent less than the 
English horse-power. Tables at the end of this book give the 
equivalents in each system of the units of the other system, but 
the student will rarely need to use such tables. He should, on 
the other hand, exclusively employ the metric system when using 
it, and learn to think readily in it. The following matter is sup¬ 
plementary to the corresponding articles of the preceding pages. 

(Art. 2 ) At about o° centigrade ice is generally formed. 
When water is kept perfectly quiet, however, it is found that its tem¬ 
perature can be reduced to — 7 0 or — 9 0 before freezing begins, but at 
this instant the temperature of the water rises to o° centigrade. 


Data in the Metric System. Art. 9 


19 


(Art. 3 ) In the metric system the following approximate values 
are used for the weight of water: 

i liter of water weighs i kilogram 
i cubic meter weighs iooo kilograms 

It may be noted that the constants for the weight of water differ 
slightly in the two systems. Thus, the equivalent of 62.5 pounds 
per cubic foot is about 1001 kilograms per cubic meter. The weight 
per unit of volume of pure distilled water is greatest at the temperature 
of maximum density, 4°.i centigrade, and least at the boiling-point. 
Table 9 a gives weights of distilled water at different temperatures 
in kilograms per cubic meter, as determined by Rossetti.* River 

Table 9a. Weight of Distilled Water 


Metric Measures 


Temperature 

Centigrade 

K.lograms per 
Cubic Meter 

Temperature 

Centigrade 

Kilograms per 
Cubic Meter 

Temperature 

Centigrade 

Kilograms per 
Cubic Meter 

- 3 ° 

999-59 

16 ° 

999.OO 

55 ° 

985-85 

0 

999.87 

18 

998.65 

60 

983-38 

+ 3 

999.99 

20 

998.26 

65 

980.74 

4 

1000.00 

22 

397-83 

70 

977-94 

5 

999.99 

25 

397.12 

75 

974.98 

6 

999.97 

30 

995-76 

80 

971-94 

8 

999.89 

35 

994-13 

85 

968.79 

10 

999-75 

40 

992.35 

90 

965-56 

12 

999-55 

45 

990.37 

95 

962.19 

14 

999.30 

50 

998.20 

100 

958.65 


waters are usually between 997 and 1001 kilograms per cubic foot, 
depending upon the amount of impurities and the temperature, while 
the water of some mineral springs has been found as high as 1004. 
It appears then that 1000 kilograms per cubic meter is a fair average 
value to use in hydraulic work for the weight of fresh water. Brack¬ 
ish and salt waters are heavier. For the Gulf of Mexico the weight 
per cubic meter is about 1023, for the oceans, about 1027, while for the 
Dead Sea there is stated the value of 1169 kilograms per cubic meter. 
For Great Salt Lake the weight of water varies from 1105 to 1227 
kilograms per cubic meter. The weight of ice per cubic meter varies 
from 916 to 921 kilograms. 

* Annales de chemie et de physique, 1869, vol. 17, p. 370. 

















20 


Chap. 1. Fundamental Data 


(Art. 4 ) Near the sea level the average reading of the mer¬ 
cury barometer is 76 centimeters, and since mercury weighs 13.6 
grams per cubic centimeter, the average atmospheric pressure is taken 
to be 76 -H 0.0136 = 1-0333 kilograms per square centimeter. One 
atmosphere of pressure is therefore slightly greater than a pressure 
of one kilogram per square centimeter. Conversely, a pressure of 
one kilogram per square centimeter may be expressed as a pressure 
of 0.968 atmosphere. In a perfect vacuum water will rise to a height 
of about io^ meters under a mean pressure of one atmosphere, for the 
average specific gravity of mercury is 13.6, and 13.6 X 0.76=10.33 
meters. Table 9 b shows atmospheric pressures, altitudes, and boil¬ 
ing-points of water corresponding to heights of the mercury and water 
barometers. 

Table %. Atmospheric Pressure 
Metric Measures 


Mercury 

Barometer 

Millimeters 

Pressure 
Kilograms 
per Square 
Centimeter 

Pressure 

Atmospheres 

Water 

Barometer 

Meters 

Elevations 

Meters 

Boiling-point 
of Water 
Centigrade 

790 

I.074 

I.04 

IO.74 

- 325 

IOI°.I 

760 

1-033 

I.OO 

10-33 

O 

IOO .0 

730 

0.992 

0.96 

9.92 

+ 340 

98 .9 

700 

•952 

.92 

9-52 

690 

97 -8 

670 

.911 

.88 

9.II 

1045 

96 .6 

640 

.870 

.84 

8.70 

1420 

95 -4 

6lO 

.829 

.80 

8.29 

1820 

94 -i 

580 

.788 

.76 

7.88 

2240 

92 .8 

550 

.748 

.72 

7.48 

2680 

9i -5 

520 

.707 

.68 

7.07 

3140 

90 .1 


(Art. 5 ) If the weight of a cubic meter of water is 1000 kilo¬ 
grams at the surface of a pond, the weight of a cubic meter at a 
depth of 10 1 meters will be 

1000(1 + 0.00005) = 1000.05 kilograms, 
and at a depth of 103^ meters a cubic meter will weigh 

1000(1 + 0.0005) = 1000.5 kilograms. 

Hence the variation due to compression is too small to be generally 
taken into account. The modulus of elasticity of volume for water is 


E = — I - . °b 3 _ = 20 700 kilograms per square centimeter, 
0.00005 5 

















Data in the Metric System. Art. 9 


21 


while that of steel is about 2 100 000. Using g = 9.8 meters per 
second per second, the mean velocity of sound in water is 

v = ^/Eg/w = 1420 meters per second. 

(Art. 6) The formula of Peirce for the acceleration of gravity 
on the earth’s surface is 

g = 9.78085 (1 + 0.0052375 sin 2 1 ) (1 - 0.000000314 e) (9)i 

in which g is the acceleration in meters per second per second at a 
place whose latitude is l degrees and whose elevation is e meters 
above the sea level. The greatest value of g is at the sea level at 
the pole; here l = go° and e = o, whence g = 9.8322. The least 
value of g in hydraulic practice is found on high lands at the 
equator ; here l = o° and e = 4000 meters, whence g = 9.7683. The 
mean of these is 9.800, which closely agrees with that found in 
Art. 6, since 32.16 feet equals 9.802 meters; accordingly 

g = 9.800 meters per second per second 

is the value of the acceleration that will be used in the metric work 
of this book. From this are found 

V2g = 4.427 1/2^ = 0.05102 (9)2 

Table 9 c gives multiples of these values which will often be of use 
in numerical computations. 


Table 9c. Acceleration Due to Gravity 

Metric Measures 


No. 

Multiples 
of g 

Multiples 
of 2 g 

Multiples 
of 1 / 2 g 

Multiples 
of 2 g 

No. 

1 

9.800 

19.60 

0.05102 

4.427 

1 

2 

19.60 

39.20 

0.1020 

8.854 

2 

3 

29.40 

58.80 

O.IS3I 

13.282 

3 

4 

39.20 

78.40 

0.2041 

17.71 

4 

S 

49.OO 

98.OO 

0.2551 

22.14 

5 

6 

58.80 

II7.60 

O.3061 

26.56 

6 

7 

68.60 

137.2 

0.3571 

30.99 

7 

8 

78.40 

156.8 

O.4082 

35-42 

8 

9 

88.20 

176.4 

O.4592 

39.84 

9 

10 

98.00 

196.O 

O.5102 

44.27 

10 


(Art. 8) The remarks as to precision of numerical computation 
also apply here. Thus, if it be required to find the weight of water 















22 


Chap. 1. Fundamental Data 


in a pipe 38 centimeters in diameter and 6 meters long, Table F gives 
0.1134 square meter for the sectional area, the volume is then 0.6804 
cubic meter, and the weight is 680 kilograms, the fourth figure 
being omitted because nothing is known about the temperature 
or purity of the water. In general, hydraulic computations are 
much easier in the metric than in the English system. 

Prob. 9 a. Compute the acceleration of gravity at Quito, Ecuador, 
which is in latitude — o 13 and at an elevation of 2850 meters above 
sea level. 

Prob. %. What is the pressure in kilograms per square centimeter 
at the base of a column of water 95.4 meters high ? 

Prob. 9 c. Compute the velocity of sound in fresh distilled water at the 
temperature of 12 0 centigrade, and also its mean velocity in salt water. 

Prob. 9 d. How many cubic meters of water are contained in a pipe 
315 meters long and 15 centimeters in diameter? How many kilograms? 
How many metric tons ? 

Prob. 9 c. What is the boiling-point of water when the mercury ba¬ 
rometer reads 735 millimeters? How high will water rise in a vacuum tube 
at a place where the boiling-point of water is 92 0 centigrade ? 


Transmission of Pressure. Art. 10 


23 


CHAPTER 2 
HYDROSTATICS 

Art. 10 . Transmission of Pressure 

One of the most remarkable properties of a fluid is its capacity 
of transmitting a pressure, applied at one point of the surface of 
a closed vessel, unchanged in intensity, in all directions, so that 
the effect of the applied pressure is to cause an equal force per 
square inch upon all parts of the enclosing surface. Pascal, in 
1646, was the first to note that great forces could be produced 
in this manner; he saw that the 
total pressure increased propor¬ 
tionally with the area of the sur¬ 
face. Taking a closed barrel filled 
with water, he inserted a small 
vertical tube of considerable length 
tightly into it, and on filling the 
tube the barrel burst under the 
great pressure thus produced on its sides, although the weight 
of the water in the tube was quite small. The first diagram in 
Fig. 10 a represents Pascal’s barrel, and it is seen that the unit- 
pressure in the water at B is due to the head AB and independent 
of the size of the tube AC. 

Pascal clearly saw that this property of water could be em¬ 
ployed in a useful manner in mechanics, but it was not until 1796 
that Bramah built the first successful hydraulic press. This 
machine has two pistons of different sizes, and a force applied to 
the small piston is transmitted through the fluid and produces 
an equal unit-pressure at every point on the large piston. The 
applied force is here multiplied to any required extent, but the 
work performed by the large piston cannot exceed that imparted 
to the fluid by the small one. Let a and A be the areas of the 


A 


c 


B 







































24 


Chap. 2. f Hydrostatics 


small and large pistons, and p the pressure in pounds per square 
unit applied to a ; then the unit-pressure in the fluid is p, and the 

total pressure on the small pis¬ 
ton is pa, while that on the large 
piston is pA. Let the distances 
through which the pistons move 
during one stroke be d and D. 
Then the imparted work is pad, 
and the performed work, neglect¬ 
ing frictional resistances, is pAD. 
Consequently ad = AD, and 
since a is small as compared with A, the distance D must be 
small compared with d. Here is found an illustration of the 
popular maxim “ What is lost in velocity is gained in force.” 

Numerous applications of this principle are made in hydraulic 
presses for compressing materials and forging steel, as also in jacks, 
accumulators, and hydraulic cranes. The Keely motor, one of the 
delusions of the nineteenth century, is said to have employed this 
principle to produce some of its effects; very small pipes, supposed by 
the spectators to be wires conveying some mysterious force, being 
used to transmit the pressure of water to a receiver where the total 
pressure became very great in consequence of greater area. 

In consequence of its fluidity the pressure existing at any 
point in a body of water is exerted in all directions with equal 
intensity. When water is confined by a bounding surface, as 
in a vessel, its pressure against that surface must be normal at 
every point, for if it were inclined, the water would move along 
the surface. When water has a free surface, the unit-pressure at 
any depth depends only on that depth and not on the shape of 
the vessel. Thus in the second diagram of Fig. 10 a the unit- 
pressure at C produced by the smaller column of water aC is the 
same as that caused by the larger column AC, and the total ver¬ 
tical pressure on the upper side of the base B is the product of 
its area into the unit-pressure caused by the depth AB. 

Prob. 10 . What is the upward pressure on the lower side of the base B 
in Fig. 10 a ? Explain why this is less than the downward pressure on the 
upper side of the base B. 


































Head and Pressure. Art. 11 


25 


Art. 11 . Head and Pressure 

The free surface of water at rest is perpendicular to the direc¬ 
tion of the force of gravity, and for bodies of water of small extent 
this surface may be regarded as a plane. Any depth below this 
plane is called a “head,” or the head upon any point is its vertical 
depth below the level surface. In Art. 10 it was seen that the 
unit-pressure at any depth depends only on the head and not on 
the shape of the vessel. Let h be the head and w the weight of 
a cubic unit of water; then at the depth h one horizontal square 
unit bears a pressure equal to the weight of a column of water 
whose height is h , and whose cross-section is one square unit, 
or wh. But the pressure at this point is exerted in all directions 
with equal intensity. The unit-pressure p at the depth h then is 
wh , and the depth, or head, for a unit-pressure p is p/w, or 

p = wh h = p/w (ll)i 

If h be expressed in feet and p in pounds per square foot, these 
formulas become, using the mean value of w, 

p = 62.$h h = o.oi6p 

Thus pressure and head are mutually convertible, and in fact 
one is often used as synonymous with the other, although really 
each is proportional to the other. Any unit-pressure p can be 
regarded as produced by a head h, which is frequently called 
the “ pressure head.” 

In engineering work p is usually taken in pounds per square 
inch, while h is expressed in feet. Thus the pressure in pounds 
per square foot is 62.5 h, and the pressure in pounds per square 
inch is 774 of this, or 

p = 0.4340 h h = 2.304 p (H)2 

These rules may be stated in words as follows: 

1 foot head corresponds to 0.434 pounds per square inch; 

1 pound per square inch corresponds to 2.304 feet head. 

These values, be it remembered, depend upon the assumption 
that 62.5 pounds is the weight of a cubic foot of water, and hence 


26 


Chap. 2 . Hydrostatics 


are liable to variation in the third significant figure (Art. 4 ). The 
extent of these variations for fresh water maybe seen in Table 11 , 
which gives multiples of the above values, and also the corre¬ 
sponding quantities when the cubic foot is taken as 62.3 pounds. 

Table 11. Heads and Pressures 


Head 
in Feet 

Pressure in Pounds per 

Square Inch 

Pressure 
in Pounds 
per 

Square 
. Inch 

Head in Feet 

■w - 62.5 

w - 62.3 

w — 62.5 

w = 62.3 

I 

0-434 

0-433 

I 

2.304 

2.311 

2 

0.868 

0.865 

2 

4.608 

4.623 

3 

1.302 

1.298 

3 

6.912 

6.934 

4 

1.736 

I - 73 I 

4 

9.216 

9.246 

5 

2.170 

2.163 

5 

II.520 

n -557 

6 

2.604 

2.596 

6 

13.824 

13.868 

7 

3-038 

3.028 

7 

16.128 

16.180 

8 

3-472 

3.461 

8 

18.432 

18.491 

9 

3.906 

3-894 

9 

20.736 

20.803 

10 

4-340 

4.326 

10 

23.040 

23.114 


The atmospheric pressure, which is about 14.7 pounds per 
square inch, is transmitted through water, and is to be added to 
the pressure due to the head whenever it is necessary to regard 
the absolute pressure. This is important in some investigations 
on the pumping of water, and in a few other cases where a partial 
or complete vacuum is produced on one side of a body of water. 
For example, if the air is exhausted from a small globe, so that 
its tension is only 6.5 pounds per square inch, and it is submerged 
in water to a depth of 250 feet, then the absolute pressure on the 
surface of the globe is 

P = °-434 X 250 + 14.7 = 123.2 pounds per square inch, 

and the resultant effective pressure on that surface is 

p' = 123.2 — 6.5 = 116.7 pounds per square inch. 

Unless otherwise stated, however, the atmospheric pressure need 
not be regarded, since under ordinary conditions it acts with 
equal intensity upon both sides of a submerged surface. 

















Loss of Weight in Water. Art. 12 


27 


Prob. 11 . How many pounds per square inch correspond to a head of 
230 feet ? How many feet head correspond to a pressure of 100 pounds per 
square inch? 

Art. 12 . Loss of Weight in Water 


It is a familiar fact that bodies submerged in water lose part 
of their weight; a man can carry under water a large stone which 
would be difficult to lift in air, and timber when submerged has 
a negative weight or tends to rise to the surface. The following 
is the law of loss which was discovered by Archimedes, about 
250 B.C., when considering the problem of King Hiero’s crown : 

The weight of a body submerged in water is less than its 
weight in air by the weight of a volume of water which is equal 
to the volume of the body. 


To demonstrate this, consider that the submerged body is 
acted upon by the water pressure in all directions, and that the 
horizontal components of these pressures must balance. Any 
vertical elementary prism is subjected to an upward pressure upon 
its base which is greater than the downward pressure upon its 
top, since these pressures are due to the 
heads. Let hi be the head on the top of 
the elementary prism and ^2 that on its 
base, and a the cross-section of the prism; 
then the downward press re is ivahi and 
the upward pressure is wall 2. The differ¬ 
ence of these, wa(Jh—h\) is the resultant 
upward water pressure, and this is equal to the weight of a 
column of water whose cross-section is a and whose height is 
that of the elementary prism. Extending this theorem to all 
the elementary prisms, it is concluded that the weight of the 
body in water is less than its weight in air by the weight of an 



equal volume of water. 

It is important to regard this loss of weight in constructions 
under water. If, for example, a dam of loose stones allows the 
water to percolate through it, its weight per cubic foot is less than 
its weight in air, so that it can be more easily moved by horizontal 
forces. As stone weighs about 150 pounds per cubic foot in air, 

















28 


Chap. 2 . Hydrostatics 


its weight in water is only about 150 — 62=88 pounds per cubic 
foot. If a cubic foot of sand, having voids amounting to 40 per 
cent of its volume, weighs no pounds, its loss of weight in water 
is 0.60 X 62.5 = 37.5 pounds, so that its weight in water is 
no —37.5 = 72.5 pounds. 

The ratio of the weight of a substance to that of an equal 
volume of water is called the specific gravity of the substance, 
and this is easily computed from the law of Archimedes after 
weighing a piece of it in air and then in water; or, if w be the 
weight of a cubic unit of water and w' the weight of a cubic unit 
of any substance, the ratio w’/w is the specific gravity of the 
substance. 

Prob. 12 . A box containing 1.17 cubic feet weighs 19.3 pounds when 
empty and 133.5 when filled with sand. It is then found that 29.7 pounds 
of water can be poured in before overflow occurs. Find the percentage 
of voids in the sand, the specific gravity of the sand mass, and the specific 
gravity of a grain of sand. 

Art. 13 . Depth of Flotation 

When a body floats upon water, it is sustained by an upward 
pressure of the water equal to its own weight, and this pressure 
is the same as the weight of the volume of water displaced by 
the body. Let W' be the weight of the floating body in air, and 
W be the weight of the displaced water; then W = W. Now 
let 2 be the depth of flotation of the body; then to find its value 
for any particular case W is to be expressed in terms of the linear 
dimensions of the body, and W in terms of the depth of flotation 2. 
For example, a timber box caisson is 20 X ioi feet in outside 
dimensions and weighs 33 400 pounds. The weight of displaced 
water in pounds is 62i X 20 X ioj X z, and equating this to 
33 400 gives 2 = 2.54 feet for the depth of flotation. 

To find the depth of flotation for a cylinder Iving horizontally, 
let w' be its weight per cubic unit, l its length, and r the radius of 
its cross-section. The depth of flotation is DE, or letting 0 be the 
angle ACE, then z = (i-cos 0 )r. The weight of the cylinder is 
W = 7rr 2 / • w\ and that of the displaced water is 

W — {r- arc 0 — r 2 sin# cos 9)1 • w 


Stability of Flotation. Art. 14 


29 


Equating the values of W and W\ and substituting for sin# cos# 
its equivalent \ sin 2 #, there results 

2 arc# — sin 2# = 2 tts 


in which 5 represents the ratio w'/w or the specific gravity of the 
material of the cylinder. From this equation # is to be found by 
trial for any particular case, and then z is computed. For example, 


if w =26.5 pounds per cubic foot, 
then 5 is 0.424, and 

2 arc# — sin 2# — 2.664 == 0 

To solve this equation, values are 
to be assumed for #, until one is 
found that satisfies it; thus from 
Table G, 



Fig. 13. 


for # = 83° 2.897 — 0.242 — 2.664 — — 0.009 

for # = 83i 2.906 — 0.234 — 2.664 = + 0.008 


Therefore # lies between 83° and 83° 15', and is probably about 
83° 8'. Hence the depth of flotation is z = (1 — o.i2o)r = o.88r, 
or if the diameter is one foot, the depth of flotation is 0.44 feet. 

In a similar way it may be shown that the depth of flotation of 
a sphere of radius r and specific gravity 5 is given by the cubic equa¬ 
tion z 3 — 3 rz 2 + 4 r 3 s = o. When r = 4 feet and s = 0.65, it may be 
found by trial that z = 1.21 feet. 

Prob. 13 . A wooden stick H inches square and 10 feet long is to be 
used for a velocity float which is to stand vertically in the water. How many 
square inches of sheet lead vs inch thick must be tacked on the sides of this 
stick so that only 4 inches will project above the water surface ? The wood 
weighs 31.25 and the lead 710 pounds per cubic foot. 


Art. 14 . Stability of Flotation 

The equilibrium of a floating body is stable when it returns 
to its primitive position after having been slightly moved there¬ 
from by extraneous forces; it is indifferent when it floats in any 
position, and it is unstable when the slightest force causes it to 
leave its position of flotation. For instance, a short cylinder 
with its axis vertical floats in stable equilibrium, but a long 
cylinder in this position is unstable, and a slight force causes it 
to fall over and float with its axis horizontal in indifferent equilib- 
























30 


Chap. 2. Hydrostatics 


rium. It is evident that the equilibrium is the more stable the 
lower the center of gravity of the body. 

The stability depends in any case upon the relative position of the 
center of gravity of the body and its center of buoyancy, the latter 
being the center of gravity of the displaced water. Thus in Fig. 14 
let G be the center of gravity of the body and let C be its center of 
buoyancy when in an upright position. Now if an extraneous force 

causes the body to tip into the posi¬ 
tion shown, the center of gravity 
remains at G, but the center of buoy¬ 
ancy moves to D. In this new posi¬ 
tion of the body it is acted upon by 
the forces W' and W, which are equal 
and parallel but opposite in direction. 
These forces form a couple which 
tends either to restore the body to the upright position or to cause it 
to deviate farther from that position. Let the vertical through D 
be produced to meet the center line CG in M. If M is above G, 
the equilibrium is stable, as the forces W and W f tend to restore it 
to its primitive position; if M coincides with G, the equilibrium is 
indifferent; and if M be below G, the equilibrium is unstable. 

The point M is called the “ metacenter,” and the theorem may be 
stated that the equilibrium is stable, indifferent, or unstable according 
as the metacenter is above, coincident with, or below the center of grav¬ 
ity of the body. The measure of the stability of a stable floating 
body is the moment of the couple formed by the forces W and W'. But 
GM is proportional to the lever arm of the couple, and hence the quan¬ 
tity W X GM may be taken as a measure of stability. The stability, 
therefore, increases with the weight of the body, and with the distance 
of the metacenter above the center of gravity. (See Art. 189 .) 

The most important application of these principles is in the design 
of ships, and usually the problems are of a complex character which can 
only be solved by tentative methods. The rolling of the ship due to 
lateral wave action must also receive attention, and for this reason 
the center of gravity should not be put too low. 

Prob. 14 . A square prism of uniform specific gravity s has the length 
h and the cross-section b 2 . When this prism is placed in water with its axis 
vertical, it may be shown that it is in stable, indifferent, or unstable equilib¬ 
rium according as b 2 is greater, equal to, or less than 6 h 2 s (i — s). 

































Normal Pressure. Art. 15 


31 


Art. 15 . Normal Pressure 


The total normal pressure on any immersed surface may be 
found by the following theorem: 

The total normal pressure is equal to the product of the 
weight of a cubic unit of water, the area of the surface, and 
the head on its center of gravity. 


To prove this let A be the area of the surface, and imagine it 
to be composed of elementary areas, ai, a^, as, etc., each of which 
is so small that the unit-pres¬ 
sure over it may be taken as 
uniform; let hi, /z 2 , h z , etc., 
be the heads on these elemen¬ 
tary areas, and let w denote 
the weight of a cubic unit of 
water. The unit-pressures at 
the depths h x , h 2 , h z , etc., are wh x , wh 2 , wh 3 , etc. (Art. 11 ), and 
hence the normal pressures on the elementary areas, a u a 2 , a 3 , etc., 
are wa Y hi, wa 2 h 2 , wa z fa, etc. The total normal pressure P on the 
entire surface then is 



Fig. 15 . 


P = w[a\h\ + a 2 h 2 + a z h z + etc.) 

Now let h be the head on the center of gravity of the surface; 
then, from the definition of the center of gravity, 

aihi + a 2 h 2 + a z h z + etc. — Ah 


Therefore the normal pressure is 

P = wAh ( 15 ) 


which proves the theorem as stated. 

This rule applies to all surfaces, whether plane, curved, or 
warped, and however they be situated with reference to the water 
surface. Thus the total normal pressure upon the surface of an 
immersed cylinder remains the same whatever be its position, 
provided the depth of the center of gravity of that surface be 
kept constant. It is best to take h in feet, A in squaie feet, and 
w as 62.5 pounds per cubic foot; then P will be in pounds. In 















32 


Chap. 2 . Hydrostatics 


case surfaces are given whose centers of gravity are difficult to 
determine, they should be divided into simpler surfaces, and then 
the total normal pressure is the sum of the normal pressures on 
the separate surfaces. 

The normal pressure on the base of a vessel filled with water 
is equal to the weight of a cylinder of water whose base is the base 
of the vessel, and whose height is the depth of water. Only in 
the case of a vertical cylinder does this become equal to the weight 
of the water, for the pressure on the base of a vessel depends upon 
the depth of water and not upon the shape of the vessel. Also 
in the case of a dam, the depth of the water and not the size of the 
pond, determines the amount of pressure. 

When a surface is plane, the total normal pressure is the result¬ 
ant of all the parallel pressures acting upon it. This is not true 
for curved surfaces ; for, as the pressures have different directions, 
their resultant is not equal to their numerical sum, but must be 
obtained by the rules for the composition of forces. For exam¬ 
ple, when a sphere of diameter d is filled with water, the total 
normal pressure as found by the formula ( 15 ) is 

P — w • 7 rdr • i d = | wird 3 

but the resultant pressure is nothing, for the elementary normal 
pressures act in all directions so that no tendency to motion 
exists. The weight of water in this sphere is J wird 3 , or one- 
third of the total normal pressure, and the direction of this is 
vertical. 

Prob. 15 . An ellipse, with major and minor axes equal to 12 and 8 
feet, is submerged so that one extremity of the major axis is 3.5 and the other 
8.5 feet below the water surface. Find the normal pressure on one side. 

Art. 16 . Pressure in a Given Direction 

The pressure against an immersed plane surface in a given 
direction may be found by obtaining the normal pressure by Art. 
15 and computing its component in the required direction, or by 
means of the following theorem : 


Pressure in a Given Direction. Art. 16 


33 


The horizontal pressure on any plane surface is equal to the 
normal pressure on its vertical projection ; the vertical pressure 
is equal to the normal pressure on its horizontal projection; and 
the pressure in any direction is equal to the normal pressure on 
a projection perpendicular to that direction. 

To prove this let A be the area of the given surface, represented 
by ^ 4^4 in Fig. 16 a, and P the normal pressure upon it, or P = wAh. 
Now let it be required to find the pres¬ 
sure P' in a direction making an angle 6 
with the normal to the given plane. 

Draw A'A' perpendicular to the direc¬ 
tion of P', and let A' be the area of 
the projection of A upon it. The 
value of P' then is 

P' = P cos 0 = wAh cos 0 

But A cos6 is the value of A' by the construction. Hence 

P' = wA'h ( 16 ) 

and the theorem is thus demonstrated. 

This theorem does not in general apply to curved surfaces. 
But in cases where the head of water is so great that the pressure 
may be regarded as uniform it is also true for curved sur¬ 
faces. For instance, consider a 
cylinder or sphere subjected on 
every elementary area to the unit- 
pressure p due to the high head h , 
and let it be required to find the 
pressure in the direction shown by 
q h q 2 , and qs in Fig. 166 . The 
pressures pi, pt, pz, etc., on the ele¬ 
mentary areas ai, a 2 , a 3 , etc., have 
the values 

pi — pah Pz = P a 2 > P* = P a 3 > etc *> 

and the components of these in the given direction are 

q l = pa i cos0i, (pi = p(h cos02, #3 = pas cos0s, etc., 




Fig. 16 a. 









34 


Chap. 2 . Hydrostatics 


whence the total pressure P' in the given direction is 

P' = p(a,i cos^i + a 2 cos# 2 + a 3 cos 63 + etc.) 

But the quantity in the parenthesis is the projection of the 
given surface upon a plane perpendicular to the given direction, 
or MN. Hence there results 

P' = p X area MN 

which is the same rule as for plane surfaces. 

For the case of a water pipe let p be the interior pressure per 
square inch, t its thickness, and d its diameter in inches. Then 
for a length of one inch the force tending to rupture the pipe 
longitudinally is pd. The tensile unit-stress S in the walls of the 
pipe acting over the area 2 1 constitutes the resisting force 2 tS. 
Since these forces are equal, it follows that 2 St = pd is the funda¬ 
mental equation for the discussion of the strength of water pipes 
under static water pressure. For example, when the tensile 
strength of cast iron is 20 000 pounds per square inch, the unit- 
pressure p required to burst a pipe 24 inches in diameter and 0.75 
inches thick is 1250 pounds per square inch, which corresponds 
to a head of 2880 feet. 

Prob. 16 . A circular plate 5 feet in diameter is immersed so that the head 
on its center is 18 feet, its plane making an angle of 30° with the vertical. 
Compute the horizontal and vertical pressures upon one side of it. 

Art. 17 . Center of Pressure on Rectangles 

The center of pressure on a surface immersed in water is the 
point of application of the resultant of all the normal pressures 
upon it. The simplest case is the following : 

When a rectangle is placed with one end in the water sur¬ 
face, the center of pressure is distant from that end two-thirds 
of the length of the rectangle. 

This theorem will be proved by the help of the graphical illus¬ 
tration shown in Fig. 17 a. The rectangle, which in practice 
might be a board, is placed with its breadth perpendicular to the 
plane of the drawing, so that AB represents its edge. It is re¬ 
quired to find the center of pressure C. For any head h the unit- 


Center of Pressure on Rectangles. Art. 17 


35 


pressure is wh (Art. 15 ), and hence the unit-pressures on one side 
of AB may be graphically represented by arrows which form a 
triangle. Now when a force P equal to the total pressure is 


applied on the other side of the 
rectangle to balance these unit- 
pressures, it must be placed 
opposite to the center of gravity 
of the triangle. Therefore AC 
equals two-thirds of A B, and the 
rule is proved. The head on C 
is evidently also two-thirds of 
the head on B. 


A A 



Another case is that shown in Fig. 175 , where the rectangle, 
whose length is B Y B 2 , is wholly immersed, the head on B l being 
a * /?i, and on B 2 being h 2 . Let 

AB l = b u AC = y, and 
AB 2 = b 2 . Now the normal 
pressure P 1 , on AB X is ap¬ 
plied at the distance I 5 X 
from A , and the normal 
pressure P 2 on AB 2 is applied at the distance § b 2 from A. The 
normal pressure P on B X B 2 is the difference of P x and P 2 , or 
P = P 2 — P v Also by taking moments about A as an axis, 

PXy = P 2 X§ 52-PiXf b x 



Now, by Art. 15 , the normal pressures P 2 and P l for a rectangle 
one unit in breadth are P 2 = \ wb 2 h 2 and Pi = i ^Mi> whence 
the total normal pressure is P — J w(b 2 h 2 — bih x ), and accordingly 
the center of pressure is given by 

_ 2 . Wh ~~ b\h\ 

3 b 2 h 2 — Mi 


When 6 is the angle of inclination of the plane to the water sur¬ 
face, the values of h 2 and hi are b 2 sin# and b\ sin#. Accord¬ 
ingly the expression becomes 



2 

3 * 


b 2 * - W* 

b 2 - b < 2 



































36 


Chap. 2. Hydrostatics 


Again, if ti is the head on the center of pressure, y — h! cosec# r 
b 2 == h 2 cosec#, and hi = hi cosec#. These inserted in the last 
equation give , 2 /z 2 3 - J h z 


h'=i 


h£_ 

J12 2 - hi 2 


( 17 ). 


These formulas are very convenient for computation, since the 
squares and cubes may be taken from tables. 


If hi equals h 2 , the above formula becomes indeterminate, 
which is due to the existence of the common factor h 2 — hi in 
both numerator and denominator of the fraction; dividing out 
this common factor, it becomes 

t / _ 2 h‘ 2 2 + 'hihi + h \ 2 
3 ’ h 2 + hi 

from which, if hi = hi = h , there is found the result Ji = h. 

Prob. 17. In Fig. 17a let the length of AB be 8.5 feet and its inclination 
to the vertical be 45 degrees. Find the depth of the center of pressure. 


Art. 18. General Rule for Center of Pressure 

For any plane surface immersed in a liquid, the center of 
pressure may be found by the following rule: 

Find the moment of inertia of the surface and its statical 
moment, both with reference to an axis situated at the intersec¬ 
tion of the plane of the surface with the water level. Divide 
the former by the latter and the quotient is the perpendicular 
distance from that axis to the center of pressure. 

The demonstration is analogous to that in the last article. 
Let B X B 2 in Fig. 17 h be the trace of the plane surface, which itself 
is perpendicular to the plane of the drawing, and C be the center 
of pressure, at a distance y from A where the plane of the surface 
intersects the water level. Let a 1} a 2 , as, etc., be elementary 
areas of the surface, and h u h 2 , /z 3 , etc., the heads upon them, 
which produce the normal elementary pressures, wa x hi, iva 2 h 2 , 
waji z , etc. Letyi, y 2 , y 3 , etc., be the distances from A to these 
elementary areas. Then taking the point A as a center of mo¬ 
ments, the definition of center of pressure gives the equation 

(' waihi + m 2 ^2 + wa 3 h 3 + etc.) y = wa\h\yi + wa^y^ + wa 3 h 3 y 3 + etc. 




General Rule for Center of Pressure. Art. 18 


37 


Now let # be the angle of inclination of the surface to the 
water level; then h x = y x sin#, Jv> = y 2 sin#, h z = y 3 sin#, etc. 
Hence, inserting these values, the expression for y is 

_ aiyi 2 + a*iy2 2 + aty * 2 + etc. 
a iyi T (hyi + #3^3 + etc. 


The numerator of this fraction is the sum of the products obtained 
by multiplying each element of the surface by the square of its 
distance from the axis, which is called the moment of inertia of 
the surface. The denominator is the sum of the products ob¬ 
tained by multiplying each element of the surface by its distance 
from the axis, which is called the statical moment of the surface. 


Therefore 


^ = 


moment of inertia _ I' 
statical moment A 


( 18 ) 


is the general rule for finding the position of the center of pressure 
of an immersed plane surface. 

The statical moment of a surface is simply its area multiplied 
by the distance of its center of gravity from the given axis. The 
moments of inertia of plane surfaces with reference to an axis 
through the center of gravity are deduced in works on theoretical 
mechanics; the following are a few values, the axis being parallel to 
the base of the rectangle or triangle: 

for a rectangle of base b and depth d, I = T2 bd 3 
for a triangle of base b and altitude d, I = y$ bd 3 
for a circle with diameter d, I = e 1 ? ^d 4 

To find from these the moment of inertia with reference to a par¬ 
allel axis, the well-known formula i' = I + Ak 2 is to be used, 
where A is the area of the surface, 
k the distance from the given axis to 
the center of gravity of the surface, 
and l' the moment of inertia re¬ 
quired. 

For example, let it be required 
to find the center of pressure of a 
vertical circle immersed so that the head on its center is 
equal to its radius. The area of the circle is } nrd 2 , and its 


















38 


Chap. 2. Hydrostatics 


statical moment with reference to the upper edge is \ ird 2 X J d. 
Then from ( 18 ) 




eV 7r ^ 4 + i ttcL 2 • i d 2 


= %d 


\ ird 2 • \ d 

or the center of pressure is at a distance § d below the center of 
the circle. 

Prob. 18. Find the depth of flotation for the triangle in Fig. 18 . Also 
find the position of the center of pressure upon it in terms of z. 


Art. 19 . Pressures on Gates and Dams 

In the case of an immersed plane the water presses equally 
upon both sides so that no disturbance of the equilibrium results 
from the pressure. But in case the water is at different levels on 
opposite sides of the surface the opposing pressures are unequal. 

For example, the cross-section of a selF 
acting tide-gate, built to drain a salt 
marsh, is shown in Fig. 19 a. On the 
ocean side there is a head of hi above 
the sill, which gives for every linear foot 
of the gate the horizontal pressure 

Pi = w X hi X \ h\ — | wit 2 

which is applied at the distance § hi 
above the sill. On the other side the 
head on the sill is h 2 , which gives the 
horizontal pressure P 2 = J wih 2 acting 
in the opposite direction to that of Pi. The resultant horizon- 
tal pressure is p _ _ p 2 _ i w (fal _ fc f) 

and if z be the distance of the point of application of P above 
the sill, the equation of moments is 

Pz = Pi X J hi P2 X ^ Jvi 

from which z can be computed. For example, if h x is 7 feet and 
h 2 is 4 feet, the resultant pressure on one linear foot of the gate 
is found to be 1031 pounds and its point of application to be 2.82 
feet above the sill. The action of this gate in resisting the water 
pressure is like that of a beam under its load, the two points of 


























Pressures on Gates and Dams. Art. 19 


39 


support being at the sill and the hinge. If h is the height of the 
gate, the reaction at the hinge is Pz/ h, and from the above expres¬ 
sion for Pz it is seen that this reaction has its greatest value when 
fa becomes equal to h and fa is zero. In the case of the vertical 
gate of a canal lock, which swings horizontally like a door, a 
similar problem arises and a similar conclusion results. 

When the water level behind a masonry dam is lower than 
its top, as in Fig. 19&, the water pressure on the back is normal 
to the plane AB and for computations this may be resolved into 




Fig. 19c. 


horizontal and vertical components. Let h be the height of water 
above the base, 0 the angle w r hich the back makes with the vertical, 
then from Arts. 15-16 the values of these pressures, for one 
linear unit of the dam, are 

Normal Pressure N = w • h sec# • \ h = \ wh 2 sec# 
Horizontal Component H = N cos# = \ wh 2 
Vertical Component V — N sin# = \ wh 2 tan# 

and from Art. 17 the point of application of these pressures is at 
a distance § h above the base. • Except in the case of hollow dams 
only the horizontal component H need usually be considered, since 
the neglect of V is on the side of safety. 

When the water runs over the top of a dam, as in Fig. 19^, 
let h be the height of the dam and d the depth of water on its 
crest. Then 

Normal Pressure N = w • h sec# • (d-\-\ h)= \ wh(h-\- 2 d) sec# 
Horizontal Component H = N cos# = J wh(h-\~ 2 d) 

Vertical Component V — N sin# = J z^(^+ 2 d)tan# 
and, from Art. 17, the point of application above the base AD is 

h + 3 d m 1 h 
















40 


Chap. 2. Hydrostatics 


when d = o, these expressions for H and p become § wh 2 and 1 h. 
If d is infinite, the value of p reduces to J h and hence in no case 
can the pressure N be applied as high as the middle of the height 
of the dam. Unless the dam be hollow or 6 be greater than 30 ° 
it will usually be proper to neglect V and to consider only H. 

It is not the place here to enter into the discussion of the subject of 
the design of masonry dams, but two ways in which they are liable to 
fail may be noted. The first is that of sliding along a horizontal joint, 
as BD ; here the horizontal component of the thrust overcomes the 
resisting force of friction acting along the joint. If W is the weight of 
masonry above the joint, and / the coefficient of friction, the resist¬ 
ing friction is fW, and the dam will slide if the horizontal component 
of the pressure is equal to or greater than this. The condition for failure 
by sliding then isH=/W. For example, consider a masonry dam of 
rectangular cross-section which is 4 feet wide and h feet high, the water 
being level with its top. Let its weight per cubic foot be 140 pounds, 
and let it be required to find the height li for which it would fail by 
sliding along the base, the coefficient of friction being 0.70. The 
horizontal water pressure is | X 62.5 X h 2 and the resisting fric¬ 
tion is 0.7 X 140 X 4 X h. Placing these equal, there is found for the 
height of the dam h = 12.5 feet. 

The second method of failure of a masonry dam is by over¬ 
turning, or by rotating about the toe D. This occurs when the moment 
of H equals the moment of W with respect to D , or if p and q are the 
lever arms dropped from D upon the directions of H and W, the condi¬ 
tion for failure by rotation is Hp=Wq. For example, when it is 
required to find the height of the above rectangular dam so that it 
will fail by rotation, the lever arms p and q are J h and 2 feet, 
and the equation of moments with respect to the toe of the dam is 

i X 62.5 X h 2 X J h = 140 X 4 X h X 2 

from which there is found ^=10.4 feet. The horizontal water 
pressure for one linear foot of the dam at the instant of failure is 
i wh 2 = 3380 pounds. 

In the case of an overfall dam, as in Fig. 19 c, the falling sheet of 
water produces a partial vacuum when air cannot freely enter behind 
it, and thus the force II, tending to produce sliding, is increased. In 
the design of a dam consideration must also be given to the upward 
pressure of that water which gains access either beneath its foundation 


Hydrostatics in Metric Measures. Art. 20 


41 


or directly into its mass. This upward pressure is equivalent to a 
loss of weight due to percolating water, as was described in Art. 12 . 


Prob. 19 . A water pipe passing through 
a masonry dam is closed by a cast-iron cir¬ 
cular valve AB , which is hinged at A, and 
which can be raised by a vertical chain BC. 

The diameter of the valve is 3 feet, its plane 
makes an angle of 27 0 with the vertical, and 
the depth of its center below the water level 
is 10.5 feet. Compute the normal water 
pressure P, and the distance of the center of 
pressure from the hinge A . Disregarding the 
weight of the valve and chain, compute the 
force F required to open the valve. When the weight of the chain is 
23 pounds and that of the valve 180 pounds, compute the force F. 



Art. 20 . Hydrostatics in Metric Measures 

(Art. 11 ) When the head h is in meters and the unit-pressure 
p is in kilograms per square meter, the formulas (ll)i become 

p = 1000/2 h = 0.001 p 

In engineering practice p is usually taken in kilograms per square 
centimeter, while h is expressed in meters. Then 

p = 0.1/2 h = 10 p (20) 

Stated in words these practical rules are: 

1 meter head corresponds to 0.1 kilogram per square centimeter 
1 kilogram per square centimeter corresponds to 10 meters head 

These values depend upon the assumption that 1000 kilograms is 
the weight of a cubic meter of water, and hence results derived from 
them are liable to an uncertainty in the third or fourth significant 
figure, as Table 20 shows. 

The atmospheric pressure of 1.033 kilograms per square centi¬ 
meter is to be.added to the pressure due to the head whenever it is 
necessary to regard the absolute pressure. For example, if the air 
is exhausted from a small globe so that its pressure is only 0.32 kilo¬ 
gram per square centimeter and it be submerged in w T ater to a depth 
of 86 meters, the absolute pressure per square centimeter on the globe 
is 0.1 X 86 -f 1.033 = 9-633 kilograms, and the resultant effective 
pressure per square centimeter is 9.633 —0.32 = 9.313 kilograms. 




















42 


Chap. 2. Hydrostatics 


Table 20. Heads and Pressures 


Metric Measures 


Head 
in Meters 

Pressure in Kilograms 
per Square Centimeter 

Pressure 
in Kilo¬ 
grams per 
Square 
Centimeter 

Head in Meters 

w — IOOO 

w = 997 

w = 1000 

w = 997 

I 

O.I 

O.0997 

I 

IO 

10.03 

2 

0.2 

O.I994 

2 

20 

20.06 

3 

0-3 

O.2991 

3 

30 

3O.O9 

4 

O.4 

O.3988 

4 

40 

40.12 

5 

o-5 

O.4985 

5 

SO 

50.15 

6 

0.6 

O.5982 

6 

60 

6o.l8 

7 

0.7 

O.6979 

7 

70 

70.21 

8 

0.8 

O.7976 

8 

80 

80.24 

9 

0.9 

O.8973 

9 

90 

90.27 

10 

1.0 

O.9970 

10 

100 

IOO.30 


(Art. 12 ) The specific gravity of a substance is expressed by 
the same number as the weight of a cubic centimeter in grams, or the 
weight of a cubic decimeter in kilograms, or the weight of a cubic 
meter in metric tons. Thus, if the specific gravity of stone is 2.4, 
a cubic meter weighs 2.4 metric tons or 2400 kilograms. A bar one 
square centimeter in cross-section and one meter long contains 100 
cubic centimeters; hence if such a bar be of steel having a specific 
gravity of 7.9, it weighs 790 grams or 0.79 kilogram in air, while 
in water it weighs 690 grams or 0.69 kilogram. 

(Art. 15 ) Here h is to be taken in meters, A in square meters, 
and w as 1000 kilograms per cubic meter; then P will be in kilograms. 

(Art. 16 ) For a water pipe let p be the interior pressure in 
kilograms per square centimeter and d its diameter in centimeters. 
Then for a length of one centimeter the force tending to rupture the 
pipe longitudinally is pd. Let S be the stress in kilograms per square 
centimeter in the walls of the pipe; this acts over the area 2/, if t 
be the thickness. As these forces are equal, the equation 2 St = pd is 
to be used for the investigation of water pipes. For example, let it 
be required to find what head will burst a cast-iron pipe 60 centime¬ 
ters in diameter and 2 centimeters thick; the tensile strength of the 
material being 1400 kilograms per square centimeter. Using the 
equation, the value of p is found to be 93.3 kilograms per square cen¬ 
timeter and then, from Art. 9 , the required head h is 933 meters. 
















Hydrostatics in Metric Measures. Art. 20 


43 


(Art. 19 ) Consider a rectangular masonry dam which weighs 
2400 kilograms per cubic meter and which is 1.4 meters thick. First, 
let it be required to find the height of water for which it would fail 
by sliding, the coefficient of friction being 0.75. The horizontal water- 
pressure is | X 1000 X h 2 , and the resisting friction is 0.75 X 2400 X 
1.4 X h; placing these equal, there is found h = 5.04 meters. Sec¬ 
ondly, to find the height for which failure will occur by rotation, the 
equation of moments is 

\ X 1000 X li 2 X J h = 2400 X 1.4 X h X 0.75 

from which there is found h = 3.89 meters. The horizontal water- 
pressure for one linear meter of this dam is \ wh 2 =*i$ 6 o kilograms. 

Prob. 20 a. In a hydrostatic press one-half of a metric horse-power is 
applied to the small piston. The diameter of the large piston is 30 centi¬ 
meters and it moves 2 centimeters per minute. Compute the pressure 
in the liquid. 

Prob. 206 . What is the specific gravity of dry hydraulic cement of 
which 20.6 cubic centimeters weigh 63.2 grams? If a cube of stone 12.4 
centimeters on each edge weighs 4.88 kilograms, what is its specific gravity ? 

Prob. 20 c. In Fig. 19 a let the head on one side of the gate be 2.5 and on 
the other side 0.6 meters above the sill. Find the resultant pressure for 
one linear meter of the gate and the distance of its point of application above 
the sill. 


44 


Chap. 3. Theoretical Hydraulics 


CHAPTER 3 

THEORETICAL HYDRAULICS 
, Art. 21. Laws of Falling Bodies 

Theoretical Hydraulics treats of the flow of water when 
unretarded by opposing forces of friction. In a perfectly smooth 
inclined trough water would flow with accelerated velocity and 
be governed by the same laws as those for a body sliding down 
a frictionless inclined plane. Such a flow is, however, never 
found in practice, for all surfaces over which water moves are 
more or less rough. Friction retards the motions caused by 
gravity so that the theoretic velocities deduced in this chapter 
constitute limits which cannot be exceeded by the actual veloc¬ 
ities. Many of the laws governing the free fall of bodies in a 
vacuum are similar to those of both theoretical and practical 
hydraulics, and hence they will here be briefly discussed. 

A body at rest above the surface of the earth immediately 
falls when its support is removed. When the fall occurs in a 
vacuum, its velocity at the end of one second is g feet, the mean 
value of g being 32.16 feet per second per second, and at the end 
of t seconds its velocity is V = gt. The distance passed through 
in the time t is the product of the mean velocity \ V by the 
number of seconds, or h = \ gt 2 . Eliminating t from these two 
equations gives 

V = V2~gli or h=V 2 /2g (21)i 

which show that the velocity varies with the square root of the 
height and that the height varies as the square of the velocity. 

When a falling body has the initial velocity u at the begin¬ 
ning of the time t, its velocity at the end of this time is V = u + gt 
and the distance passed over in that time is h = ut + \ gt 2 . 
Eliminating t from these equations gives 

V = Vigh + M 2 or h=(V 2 -u 2 )/2g (21). 




Laws of Falling Bodies. Art. 21 


45 


as the relations between V and h for this case. These formulas 
are also true whatever be the direction of the initial velocity u. 

When a body of weight W is at the height h above a given 
horizontal plane, its potential energy with respect to this plane 
is Wh. When it falls from rest to this plane, the potential energy 
is changed into the kinetic energy WV 2 /2g if no work has 
been done against frictional resistance, and therefore V 2 = 2 gh. 
When it has a velocity u in any direction at the height h above 
the plane, its energy there is partly potential and partly kinetic, 
the sum of these being Wh + W * u 2 /2g\ on reaching the plane 
it has the kinetic energy WV 2 /2g. Placing these equal, there 
results V 2 = 2gh + u 2 , as found above by another method. In 
general, reasoning from the standpoint 
of energy is more satisfactory than 
that in which the element of time is 
employed. 

The general case of a body moving 
toward the earth is represented in 
Fig. 21 . When the body is at A, it is 
at a height h x above a certain horizontal plane and has the 
velocity v x . When it has arrived at B, its height above the 
plane is h 2 and its velocity is v 2 . In the first position the sum 
of its potential and kinetic energy with respect to the given 
horizontal plane is . 2 \ 

w(h+ v A 

\ 2 gJ 

and in the second position the sum of these energies is 

w(lk + ^) 

\ 2 gJ 

If no energy has been lost between the two positions, these two 
expressions are equal, and hence 

/*! + —= + — ( 21)3 

2 g 2 g 

This equation is the simplest form of Bernouilli’s theorem (Art. 
31 ). It contains two heights and two velocities, and when 


A 



Fig. 21. 




46 


Chap. 3. Theoretical Hydraulics 


three of these quantities are given, the fourth can be found ; 
thus, if hi, and Jh are given, the value of v 2 is 

V 2 = V 2 g(/Zi — h 2 )+Vi 2 

where h x — is the vertical height of A above B. With proper 
changes in notation this expression reduces to (21) 2 , which is for 
the case where the horizontal plane passes through B, and to ( 21 )!, 
which is the case where there is no initial velocity. 

Prob. 21 . A body enters a room through the ceiling with a velocity of 47 
feet per second, and in a direction making an angle of 17° with the ver¬ 
tical. If the height of the room is 16 feet, find the velocity of the body 
as it strikes the floor, resistances of the air being neglected. 

Art. 22 . Velocity of Flow from Orifices 

When an orifice is opened, either in the base or side of a vessel 
containing water, the water flows out with a velocity which is 
greater for high heads than for low heads. The theoretic velocity 
of flow is given by the theorem established by Torricelli in 1644 : 

The theoretic velocity of flow from the orifice is the 
same as that acquired by a body after having fallen from 
rest in a vacuum through a height equal to the head of 
water on the orifice. 

One proof of this theorem is by experience. When a vessel is 
arranged, as in the first diagram of Fig. 22 , so that a jet of water 
from an orifice is directed vertically upward, it is known that it 

never attains to the height of 
the level of the water in the 
vessel, although under favor¬ 
able conditions it nearly reaches 
that level. It may hence be 
inferred that the jet would 
actually rise to that height 
were it not for the resistance 
of the air and the friction of 
the edges of the orifice. Now, since the velocity required to 
raise a body vertically to a certain height is the same as that 
acquired by it in falling from rest through that height, it is re- 





















Velocity of Flow from Orifices. Art. 22 


47 


garded as established that the velocity at the orifice is that stated 
in the theorem. 

The following proof rests on the law of conservation of energy. 
Let, as in the second diagram of Fig. 22 , the water surface in a vessel 
be at A and let the flow through the orifice occur for a very short in¬ 
terval of time during which the water surface descends to A\. Let 
W be the weight of water between the planes A and A 1} which is evi¬ 
dently the same as that which flows from the orifice during the short 
time considered. Let Wi be the weight of water between the planes 
A i and B , and hi the height of its center of gravity above the orifice. 
Let h be the height of A above the orifice, and Sh the small distance 
between A and A±. At the beginning of the flow the water in the vessel 
has the potential energy Wihi-j-W (h—^Sh) with respect to B. 
V being the velocity at the orifice, the same water at the end of the 
short interval of time has the energy WihiA~W • V 2 1 2g. By the law 
of conservation these are equal if no energy has been expended in 
overcoming frictional resistances ; thus h—\ 8h = V 2 / 2g. Here 8 h 
is very small if the area A is large compared with the area of the ori¬ 
fice, and thus V 2 = 2gh , which is the same as for a body falling from 
rest through the height h. Or h—\8h may be regarded as an aver¬ 
age head corresponding to an average velocity V, so that in general 
V 2 / 2g is equal to the average head on the orifice. 

For any orifice, therefore, whether its plane is horizontal, 
vertical, or inclined, provided the head h is so large that it has 
practically the same value for all parts of the orifice, the relation 
between V and h is 

V = ^J~2gh or li = V 2 / 2 g ( 22 )i 

the first of which gives the theoretic velocity of flow due to a given 
head, while the second gives the theoretic head that will produce 
a given velocity. The term “ velocity-head ” will generally be 
used to designate the expression V 2 / 2 g, this being the height to 
which the jet would rise if it were directed vertically upward and 
there were no frictional resistances. Using for g the mean value 
32.16 feet per second per second (Art. 7 ), these formulas become 

V = 8.020 V/j h = 0.01555 V 2 (22)2 

in which h must be in feet and V in feet per second. The follow¬ 
ing table gives values of the velocity V corresponding to a given 



48 


Chap. 3. Theoretical Hydraulics 


head h and also values of the velocity-head h corresponding to a 
given velocity V. It is seen that small heads produce high theo¬ 
retic velocities. The relation between h and V is the same as that 
between the ordinate and abscissa of the common parabola when 
the origin is at the vertex. It may also be noted that the dis¬ 
cussion here given applies not only to water but to any liquid; 
thus V 2 = 2gh is theoretically true for alcohol and mercury as 
well as for water. 


Table 22. Velocities and Velocity-heads 



V = V2f A 

= 8.020 ~\fh 


h = FV2 g = 

0 .01555 V 2 

Head 
iu Feet 

Velocity 
in Feet 
per Second 

Head 
in Feet 

Velocity 
in Feet 
per Second 

Velocity 
in Feet 
per 

Second 

Head 
in Feet 

Velocity 
in Feet 
per 

Second 

Head 
in Feet 

O.I 

2-537 

I 

8.02 

I 

O.Ol6 

IO 

1.56 

0.2 

3.587 

2 

H -33 

2 

0.062 

20 

6.22 

0-3 

4-393 

3 

13.89 

3 

0.140 

30 

13-99 

0.4 

5.072 

4 

16.04 

4 

O.249 

40 

24.88 

o-S 

5-671 

5 

17-93 

5 

0.389 

50 

38.87 

0.6 

6.212 

6 

19.64 

6 

0.560 

60 

55-97 

0.7 

6.710 

7 

21.22 

7 

0.762 

70 

76.19 

0.8 

7 .i 7 i 

8 

22.68 

8 

0-995 

80 

99 - 5 1 

0.9 

7.608 

9 

24.06 

9 

I.260 

90 

I2 5-95 

1.0 

8.020 

10 

25.36 

10 

i -555 

IOO 

i 55 . 5 o 


When a Pitot tube (Art. 41 ) is placed with its mouth in the 
plane of the horizontal orifice in Fig. 22 , and at the contracted 
section of the jet (Art. 45 ), it will be found that the water in 
it stands practically at the level of the water in the vessel.* In 
this manner the frictional resistance of the air is eliminated, 
and a valuable experimental demonstration of the theorem 
which connects the velocity and the velocity-head is 
obtained. 

Prob. 22 . Find from Table 22 the velocity due to a head of 0.085 
feet, and the velocity-head corresponding to a velocity of 65.5 feet per 
second. 


* Engineering Record, Feb. 15, 1902. 
























// 


Flow under Pressure. Art. 23 


49 


Art. 23 . Flow under Pressure 

The level of water in the reservoir and the orifice of outflow 
have been thus far regarded as subjected to no pressure, or at least 
only to the pressure of the atmosphere which acts upon both with 
the same mean force of 14.7 pounds per square inch, since the 
head h is rarely or never so great that a sensible variation in at¬ 
mospheric pressure can be detected between the orifice and the 
water level. But the upper level of the water may be subject to 
the pressure of steam or to the pressure due to a heavy weight or 
to a piston. The orifice may also be under a pressure greater or 
less than that of the Itmosphere. It is required to determine 
the velocity of flow from the orifice under these conditions. 

First, suppose that the surface of the water in the vessel or 
reservoir is subjected to the uniform pressure of p Q pounds per 
square unit above the atmospheric pressure, while the pressure 
at the orifice is the same as that of the atmosphere. Let h be 
the depth of water on the orifice. The velocity of flow V is greater 
than V2 gh on account of the pressure p Q , and it is evidently the 
same as that from a column of water whose height is such as to 
produce the same pressure at the orifice. If w is the weight of 
a cubic unit of water, the unit-pressure at the orifice due to the 
head is wh , and the total unit-pressure at the depth of the orifice 
is p = wh + p 0 , and from formula (ll)i the head of water which 
would produce this total unit-pressure is 

t=h+^ 

w w 

Accordingly the theoretic velocity of flow from the orifice is 

V = V 2g(h + p 0 /w) 

or, if h 0 denote the head corresponding to the pressure p 0 , 

V = V2 g(h+ ho) 

The general formula ( 22 ) x thus applies to any small orifice if II 
be the head corresponding to the static pressure at the orifice. 

Secondly, suppose that the surface of the water in the vessel 
is subjected to the unit-pressure p Q , while the orifice is under the 





50 


Chap. 3. Theoretical Hydraulics 


external unit-pressure p x . Let h be the head of actual water on 
the orifice, h 0 the head of water which will produce the pressure 
p 0 , and hi the head which will produce p lm The theoretic ve¬ 
locity of flow at the orifice is then the same as if the orifice were 
under a head h + h Q — h x , or 

V = V 2 g(h + h 0 - h) (23)i 

in which the values of ho and hi are 

ho = po/w and hi = pi/w 

Usually po and pi are given in pounds per square inch, while 
ho and hi are required in feet; then (Art. 11 ) 

ho = 2.304/^0 hi = 2.304 pi 

The values of p 0 and pi may be absolute pressures, or merely pres¬ 
sures above the atmosphere. In the latter case pi may sometimes 
be negative, as in the discharge of water into a condenser. 

As an illustration of these principles let the cylindrical tank 
in Fig. 23 be 2 feet in diameter, and upon the surface of the water 

let there be a tightly fitting pis¬ 
ton which with the load W weighs 
3000 pounds. At the depth 8 feet 
below the water level are three 
small orifices: one at A , upon 
which there is an exterior head of 
water of 3 feet; one not shown 
in the figure, which discharges 
directly into the atmosphere; and 
one at C, where the discharge is 
into a vessel in which the air pressure is only 10 pounds per 
square inch. It is required to determine the velocity of efflux 
from each orifice. The head ho corresponding to the pressure on 
the upper water surface is 

ho = — =- 3222 -= 15.28 feet 

w 3.142X62.5 

The head hi is 3 feet for the first orifice, o for the second, and 
- 2.304(14.7 - 10) =-10.83 feet for the third. The three 
theoretic velocities of outflow then are: 


































Influence of Velocity of Approach. Art. 24 


51 


V = 8.02 V8 + 15.28 — 3 = 36.1 feet per second, 

V = 8.02 V8 + 15.28 + o = 38.7 feet per second, 

V = 8.02 V8 + 15.28 + 10.83 = 4-6-8 feet per second. 

In the case of discharge from an orifice under water, as at A 
in Fig. 23 , the value of h — h x is the same wherever the orifice be 
placed below the lower level, and hence the velocity depends 
upon the difference of level of the two water surfaces, and not 
upon the depth of the orifice. 

The velocity of flow of oil or mercury under pressure is to be de¬ 
termined in the same manner as water by finding the heads which will 
produce the given pressure. Thus in the preceding numerical example, 
if the liquid is mercury whose weight per cubic foot is 850 pounds 
the head of mercury corresponding to the pressure of the piston is 

Jiq =- 3222 -=1.12 feet, 

3.142 X 850 

and, accordingly, for discharge into the atmosphere at the depth 
h = 8 feet the velocity is 

V = 8.02 Vs + 1.12 = 24.2 feet per second, 

while for water the velocity was 38.7 feet per second. The general 
formula (22) 1 is applicable to all cases of the flow of liquids from 
a small orifice if for h its value p/w be substituted where p is the re¬ 
sultant unit-pressure at the depth of the orifice and w the weight of 
a cubic unit of the liquid. Thus for any liquid 

F = V2 gp/w (23)2 

is the theoretic velocity of flow from the orifice. Accordingly for the 
same unit-pressure p the velocities are inversely proportional to the 
square roots of the densities of the liquids. 

Prob. 23 . What is the theoretic velocity of flow from a small 
orifice in a boiler 1 foot below the water level when the steam-gage reads 
60 pounds per square inch ? What is the theoretic velocity when the 
gage reads o ? 

Art. 24 . Influence of Velocity of Approach 

Thus far in the determination of the theoretic velocity and 
discharge from an orifice, the head upon it has been regarded 
as constant. But if the cross-section of the vessel is not large, 








52 


Chap. 3. Theoretical Hydraulics 


the head can only be kept constant by an inflow of water, and this 
will modify the previous formulas. In this case the water ap¬ 
proaches the orifice with an initial velocity. Let a be the area of 
the orifice and A the area of the horizontal cross-section of the 

vessel. Let V be the velocity of flow through 
a and v be the vertical velocity of inflow 
through A. Let W be the weight of water 
flowing from the orifice in one second; then 
an equal weight must enter at A in one sec¬ 
ond in order to maintain a constant head h. 
The kinetic energy of the outflowing water is 
W • V 2 / 2 g, and this is equal, if there be no loss of energy, to 
the potential energy Wh of the inflowing water plus its kinetic 
energy W • v 2 / 2 g, 



Fig. 24a. 


or 


V 2 ir 

W v — = Wh+W- 


ig 


2g 


Now since the same quantity of water Q passes through the two 
areas in one second, Q = aV = Av, whence v = V • a/A. In¬ 
serting this value of v in the equation of energy, there is found 


V = 


2 e!l 


I — (a/AY 


( 24 ), 


which is always greater than the value V2 gh. 

The influence of the velocity of approach on the velocity of 
flow at the orifice can now be ascertained by assigning values to 
the ratio a/A. Thus, if a = A, the velocity V must be infinite 
in order that the water may fill the entire section of the vessel 


Further, 

for a = | A 

V = I.342 ^ 2 gh 

for 

cl 2 A 

V= 1.154 V2 gh 

for 

a = § A 

V = 1. 061 V2 gh 

for 

ih[l <5 

1 ! 

<3 

V = 1.02 1 V 2 gh 

for 

a ~ to ^4 

V = 1.005 v 2 gh 


It is here seen that the common formula ( 22 ) 1 is in error 2.1 per¬ 
cent when a = J A, if the head be maintained constant by a uni- 
























Influence of Velocity of Approach. Art. 24 


53 


form vertical inflow at the water surface, and 0.5 percent when 
a = to A. Practically, if the area of the orifice be less than one- 
twentieth of the cross-section of the vessel, the error in using the 
formula V = V2 gh is too small to be noticed, even in the most 
precise experiments, and fortunately most orifices are smaller in 
relative size than this. 

« 

A more common case is that where the reservoir is of large 
horizontal and small vertical cross-section, and where the water 
approaches the orifice with velocity in a horizontal direction, as 
in Fig. 24 b. Here let A be the area of the vertical cross-section 
of the trough or pipe, a the area of the orifice, and h the head on 
its center. Then if h be large compared with the depth of the 



Fig. 24 b. Fig. 24c. 


orifice, exactly the same reasoning applies as before, and the 
theoretic velocity at the orifice is given by the above formula ( 24 ) 1. 
The same is also true for the case shown in Fig. 24 c, where water 
is forced through a hose with the velocity v and issues from a 
nozzle with the velocity V, the head h being that due to the pres¬ 
sure at the entrance of the nozzle. 

The “effective head” on an orifice is the head that will pro¬ 
duce the theoretic velocity V. If H is this effective head, then 
H = V 2 /2g, and from the first equation of this article 

H = h + ~ ( 24) 2 

The effective head on an orifice is, therefore, the sum of the 
pressure and velocity heads which exist behind it. Another 
expression for the effective head can be obtained from ( 24 )i, or 


i-(o/AY 
































54 


Chap. 3. Theoretical Hydraulics 


When H has been found from either of these formulas, the 
theoretic velocity and discharge are given by 

V = V2 gH and Q = a V = a V2 gH 

for all instances where h is sufficiently large so that its value is 
sensibly constant for all parts of the orifice. But if this is not 
the case, the value of Q is to be found by the methods of Arts. 
47 and 48 . 

Prob. 24 . In Fig. 24 c let the head h be 50 feet, the diameter of the nozzle 
i\ inches, and the diameter of the hose 3 inches. Compute the effective 
head H, and also the discharge Q in cubic feet per second. 

Art. 25. The Path of a Jet 

When a jet of water issues from a small orifice in the vertical 
side of a vessel or reservoir, its direction at first is horizontal, but 
the force of gravity immediately causes the jet to move in a curve 
which will be shown to be the common parabola. Let x be the 

abscissa and y the ordinate of any 
point of the curve, measured from the 
orifice as an origin, as seen in Fig. 2 oa. 
The effect of the impulse at the orifice 
is to cause the space x to be described 
uniformly in a certain time /, or, if v be 
the velocity of flow, x = vt. The effect 
of the force of gravity is to cause the 
space y to be described in accordance 
with the laws of falling bodies (Art. 21), or y = \gt 2 . Elimi¬ 
nating t from these two equations, and replacing v 2 by its 
theoretic value 2 gh, gives 

y = gx 2 / 2v 2 = x 2 / \h 

which is the equation of a parabola whose axis is vertical and 
whose vertex is at the orifice. 

The horizontal range of the jet for any given ordinate y is 
found from the equation x 2 = 4hy. If the height of the vessel 
be /, the horizontal range on the plane of the base is 



x = 2 V/z(/ — ~h) 






















The Path of a Jet. Art. 25 


55 


This value is o when h = o and also when h = /, and it is a maxi¬ 
mum when h — hi. Hence the greatest range is from an orifice 
at the mid-height of the vessel. 

A more general case is that where the side of the vessel is 
inclined to the vertical at the angle 6, as in Fig. 25 b. Here the 
jet at first issues perpendicularly 
to the side with a velocity v, 
having the theoretic value V2 gh, 
and under the action of the im¬ 
pulsive force a particle of water 
would describe the distance AB 
in a certain time t with the uni¬ 
form velocity v. But in that 
same time the force of gravity causes it to descend through the 
distance BC. Now let x be the horizontal abscissa and y the ver¬ 
tical ordinate of the point C measured from the origin A. Then 
AB = x sec#, and BC = x tan# — y. Hence 

x sec 0 = vt x tan 0 — y = \ gt 2 

The elimination of t from these expressions gives, after replac¬ 
ing v 2 by its value 2 gh, 

y = x tan# — x 2 sec 2 #/^// ( 25 ) 

which is also the equation of a common parabola. 

To find the horizontal range in the level of the orifice take 
y = o in the last equation ; then 

x = 4 li tan#/sec 2 # = 2 h sin 26 

This is o when # = o° or # = 90°; it is a maximum and equal 
to 2 h when 9 = 45 0 . To find the highest point of the jet the 
first derivative of y with reference to x is to be equated to zero 
in order to obtain the maximum ordinate, and there results 

x — h sin 2 6 y — h sin 2 # 

which are the coordinates of the highest point with respect to 
the origin A . In these if # = 90°, x is o and y is h ; that is, if a 
jet be directed vertically upward, it will, theoretically, rise to the 
height of the water level in the reservoir. 

















56 


Chap. 3. Theoretical Hydraulics 


As a numerical example let a vessel whose height is 16 feet 
stand upon a horizontal plane DE, Fig. 2 5 b, the side of the vessel 
being inclined to the vertical at the angle 6 = 30°. Let a jet 
issue from a small orifice at A under a head of 10 feet. The jet 
rises to its maximum height, y = \ X 10 = 2.5 feet, at the dis¬ 
tance x = JV3 X 10 = 8.66 feet from A. At x = 17.32 feet 
the jet crosses the horizontal plane through the orifice. To locate 
the point where it strikes the plane DE , the value of y is made —6 
feet; then, from the equation of the curve, x is found to be 24.6 
feet, whence the distance DE is 21.2 feet. 

In practice the above equations are modified by the frictional 
resistance of the edges of the orifice which renders v less than the 
theoretic value V2 gh, and also by the resistance of the air. 
They are, indeed, extreme limits which may be approached but 
not reached by equations that take these resistances into account. 

Prob. 25. A jet issues from a vessel under a head of 6 feet, one side 
of the vessel being inclined to the vertical at an angle of 45 0 and its depth 
being 10 feet. Find the maximum height to which the jet rises, the point 
where it strikes the horizontal plane of the base, and its theoretic velocity as 
it strikes that plane. 

Art. 26. The Energy of a Jet 

Let a jet or stream of water have the velocity v, and let W be 
the weight of water per second passing any given cross-section. 
The kinetic energy of this moving water is the same as that stored 
up by a body of weight W falling freely under the action of gravity 
through a height h and thereby acquiring the velocity v. Thus, 
if K represents kinetic energy per second, 

K = Wh = W • v 2 /2 g ( 26 )i 

Now if a be the area of the cross-section and w the weight of a 
cubic unit of water, W is the weight of a prism of water of 
length v and cross-section a, or W = wav, whence 

K = wav z / 2 g ( 26) 2 

and accordingly the energy which a jet can yield in one second 
is directly proportional to its cross-section and to the cube of its 
velocity. The term “power” is often used to express energy 



The Energy of a Jet. Art. 26 


57 


per second, and when K is in foot-pounds per second, the horse¬ 
power that a jet can yield is ascertained by dividing K by 550. 
Hence the horse-powers of jets of the same cross-section vary 
as the cubes of their velocities. For example, if the velocity of a 
jet be doubled, the cross-section remaining the same, the horse¬ 
power is made eight times as great. The term “ energy of a 
jet ” is often used in hydraulics for brevity, but it always means 
energy per second of the jet; that is, the power of the jet. 

The expressions just deduced give the theoretic energy of the 
jet, that is, the maximum work which can be obtained from it in 
one second, but this, in practice, can never be fully utilized. The 
actual work realized when a jet strikes a moving surface, like the 
vane of a water-motor, depends upon a number of circumstances 
which will be explained in a later chapter, and it is the constant 
aim of inventors so to arrange the conditions that the work real¬ 
ized may be as near the theoretic energy as possible. The “ effi¬ 
ciency ” of an apparatus for utilizing the power of moving water 
is the ratio of the work k actually utilized to the theoretic energy, 

or the efficiency e is , 

e=k/K (2b) 3 

The greatest possible value of e is unity, but this can never be 
attained, owing to the imperfections of the apparatus and the 
frictional resistances. Values greater than 0.90 have, however, 
been obtained ; that is, 90 percent or more of the theoretic power 
of the water has been utilized in some of the best forms of hy¬ 
draulic motors. 

For example, let water issue from a pipe 2 inches in diameter 
with a velocity of 10 feet per second. The cross-section in square 
feet is 3.142/144, and the kinetic energy of the jet in foot-pounds 
per second is 

K = 0.01555 X 62.5 X 0.0218 X io ;5 = 21.2 

which is 0.0385 horse-power. If the velocity is 100 feet per sec¬ 
ond, the theoretic horse-power will be 38.5 ; if this jet operates 
a motor yielding 27.7 effective horse-powers, the efficiency of the 
apparatus is 27.7/38.5 = 0.72, or 72 percent of the theoretic 
energy is utilized. 


58 


Chap. 3. Theoretical Hydraulics 


The energy of a jet is the same whether its direction of motion 
be vertical, horizontal, or inclined, and per second it is always IT//, 
where h is the velocity-head corresponding to actual velocity v, 
and W is the weight of water delivered per second. The energy 
should not be computed from the theoretical velocity V, as this 
is usually greater than the actual velocity. 

J Prob. 26 . When water issues from a pipe with a velocity of 3 feet per 
second, its kinetic energy is sufficient to generate 1.3 horse-powers. What is 
the horse-power when the velocity becomes 6 feet per second ? 

Art. 27. Impulse and Reaction of a Jet 

When a stream or jet is in motion, delivering W pounds of 
water per second with the uniform velocity v, that motion may 
be regarded as produced by a constant force F, which has acted 
upon W for one second and then ceased. In this second the 
velocity of W has increased from o to v, and the space \ v has been 
described. Consequently the work F X iv has been imparted to 
the water by the force F. But the kinetic energy of the moving 
water is W • v 2 /2g, and hence by the law of conservation of energy 
F X 2 v = W X v 2 / 2 g, from which the constant force is 

F = W . v/g ( 27 )i 

This value of F is called the “impulse” of the jet. As W is in 
pounds per second, v in feet per second, and g in feet per second 
per second, the value of F is in pounds. 

In theoretical mechanics, the term “impulse” is used in a 
slightly different sense, namely, as force multiplied by time. 
In hydraulics, however, W is not pounds, but pounds per second, 
and thus the impulse is simply pounds. The force F is to be re¬ 
garded as a continuous impulsive pressure acting at the origin 
of the jet in the direction of the motion. For, by the definition, 
F acts for one second upon the W pounds of water which pass a 
given section; but in the next second W pounds also pass, and 
the same is the case for each second following. This impulse 
will be exerted as a pressure upon any surface which is placed in 
the path of the jet. 


Impulse and Reaction of a Jet. Art. 27 59 

The reaction of a jet upon a vessel occurs when water hows 
from an orifice. This reaction must be equal in value and oppo¬ 
site in direction to the impulse, as in all cases of stress action 
and reaction are equal. In the direction of the jet the impulse 
produces motion, in the opposite direction it produces an equal 
pressure which tends to move the vessel backward. The force 
of reaction of a jet is hence equal to the impulse but opposite in 
direction. For example (Fig. 27), let a vessel 
containing water be suspended at A so that 
it can swing freely, and let an orifice be 
opened in its side at B. The head of water at 
B causes a pressure which acts toward the left 
and causes W pounds of water to move during 
every second with the velocity of v feet per 
second, and which also acts toward the right and causes the 
vessel to swing out of the vertical; the first of these forces is 
the impulse, and the second is the reaction of the jet. If a force 
R be applied on the right of a vessel so as to prevent the swing- 
ing, its value is R = p = W . v / g (27), 

and this is the formula for the reaction of the jet. 

The impulse or reaction of a jet issuing from an orifice is 
double the hydrostatic pressure on the area of the orifice. Let 
h be the head of water, a the area of the orifice, and w the weight 
of a cubic unit of water; then, by Art. 15, the normal pressure 
when the orifice is closed is wah. When the orifice is opened, the 
weight of water issuing per second is W = wav, and hence the 
impulse or reaction of the jet is 

R = F = wav • v/g = 2 wa • v 2 / 2 g = 2 wah 

which is double the hydrostatic pressure. This theoretic con¬ 
clusion has been verified by many experiments (Art. 153). 

When a jet impinges normally on a plane, it produces a dynamic 
pressure on that plane equal to the impulse F, since the force re¬ 
quired to stop W pounds of water in one second is the same as 
that required to put it in motion. Again, if a stream moving 
with the velocity v is retarded so that its velocity becomes v 2 , 











60 


Chap. 3. Theoretical Hydraulics. 


the impulse in the first instant is IF • i\/g, and in the second 
W • v 2 /g . The difference of these, or 

Fi - F 2 = W (vi -v 2 )/g (27) 3 

is a measure of the dynamic pressure which has been developed. 
It is by virtue of the pressure due to change of velocity that tur¬ 
bine wheels and other hydraulic motors transform the kinetic 
energy of moving water into useful work. 

Prob. 27. If a stream of water 3 inches in diameter issues from an orifice 
in a direction inclined downward 26° to the horizon with a velocity of 15 
feet per second, find its horizontal reaction on the vessel. 


Art. 28. Absolute and Relative Velocities 

Absolute velocity is defined in this book as that with respect 
to the surface of the earth, and relative velocity as that with 
respect to a body moving on the earth. Thus absolute velocity 
is that seen by a spectator who is on the earth, and relative veloc¬ 
ity is that seen by one who is on the moving body. For instance, 
if a body is dropped by a person who is on a moving railroad car, 
it appears to a person standing outside to move obliquely, but to 
one on the car it appears to move vertically. On a car in uni¬ 
form motion all the laws of mechanics prevail exactly as if it were 
at rest; hence if a body of weight IF is dropped through a height 
h , it acquires a theoretic vertical velocity of V 2 gh with respect to 
the car. But if the horizontal velocity of the car is u, the kinetic 
energy of the body at the moment of letting it fall is IF • w 2 / 2 g 
and its potential energy is IF/?, so that, neglecting frictional re¬ 
sistances, its total energy after falling through the height h is the 
sum of these, and accordingly its absolute velocity with respect 
to the earth is V 2 gh + u 2 . 

When a vessel containing water with a free surface, as in Fig. 
28a, has an orifice under the head h and is in motion in a straight 
line with the uniform absolute velocity u, the theoretic velocity 
of flow relative to the vessel is V = V 2 gfi, or the same as its 
absolute velocity if the vessel were at rest, for no accelerating 
forces exist to change the direction or the value of g. The abso- 





Absolute and Relative Velocities. Art. 28 


61 


lute velocity of flow, however, may be greater or less than F, 
depending upon the value of u and its direction. To illustrate, 
take the case of a vessel in 
uniform horizontal motion from 
which water is flowing through 
three orifices. At A the direc¬ 
tion of F is horizontal, and as 
the vessel is moving in the op¬ 
posite direction with the velocity 
u, the absolute velocity of the water as it leaves the orifice is 
v = V — u. It is also plain, if the orifice is in front of the vessel 
and the direction of F is horizontal, that the absolute velocity of 
the water as it leaves the orifice is v = V + a. 



Again, at B is an orifice from which the water issues vertically 
with respect to the vessel with the relative velocity F, while at 
the same time the orifice moves horizontally with the absolute 
velocity u. Forming the parallelogram, the absolute velocity v 
is seen to be the resultant of the velocities F and w, or 

v = V F 2 + u 2 


Lastly, at C is shown an orifice in the front of the vessel so ar¬ 
ranged that the direction of the relative velocity F makes an 
angle </> with the horizontal. From C draw Cn to represent 
the velocity u, and CV to represent F, and complete the paral¬ 
lelogram as shown; then Cv, the resultant of u and F, is the 
absolute velocity with which the water leaves the orifice. From 
the triangle Cuv 

v = VF 2 + u 2 T- 2 uV cos </> (28) 

In this, if <£> = o, the absolute velocity v becomes F + n, as before 
shown for an orifice in the front; if $ = 90 °, it becomes the same 
as when the water issues vertically from the orifice in the base; 
and if </> = 180 °, the value of v is F — u as before found for an „ 
orifice in the rear end. 

Another case is that of a revolving vessel having an opening 
from which the water issues horizontally with the relative velocity 
F, while the orifice is moving horizontally with the absolute 




















62 


Chap. 3. Theoretical Hydraulics 


velocity u. Fig. 28 b shows this case, 0 being the angle which V 
makes with the reverse direction of u , and here also 


v = v/F 2 + u 2 — 2 uV cos /3 

is the absolute velocity of the water as it leaves the vessel. In 
all cases the absolute velocity of a body leaving a moving surface 

is the diagonal of a parallelogram, one side 
of which is the velocity of the body relative 
to the surface and the other side is the 
absolute velocity of that surface. 

When a vessel moves with a motion 
which is accelerated or retarded, this 
affects the value of g, and the reasoning 
of the preceding articles does not give the 
correct value of V. For instance, when a vessel moves verti¬ 
cally upward with an acceleration /, the relative velocity of 
flow from an orifice in it is V = V 2 (g +/) h, and if u be the 
velocity of the vessel at any instant, the absolute downward 
velocity of flow is V — u. Again, when it moves downward 
with the acceleration /, the relative velocity of flow is 
V = V2 (g —f)h and the absolute is V + u. If the downward 
acceleration is g, the vessel is freely falling and V will be zero, 
since both vessel and water are alike accelerated and there is 
then no pressure on the base. 



Prob. 28 . I11 Fig. 28 a let the orifice at A be under a head of 5.5 feet 

and its height above the earth be 7.5 feet, while the car moves with a 
velocity of 40 miles per hour. Compute the relative velocity V, the 
absolute velocity v, and the absolute velocity of the jet as it strikes 
the earth. 


Art. 29. Flow from a Revolving Vessel 

Water in a vessel at rest on the surface of the earth is acted 
upon only by the vertical force of gravity, and hence its surface 
is a horizontal plane. Water in a revolving vessel is acted upon 
by centrifugal force as well as by gravity, and it is observed that 
its surface assumes a curved shape. The simplest case is that of 
a cylindrical vessel rotating with uniform velocity about its 






Flow from a Revolving Vessel. Art. 29 


63 


vertical axis, and it will be shown that here the water surface 
is that of a paraboloid. 

Let BC be the vertical axis of the vessel, h the depth of water 
in it when at rest, and hi and h 2 the least and greatest depths of 
water in it when in motion. Let G be any point on the surface 
of the water at the horizontal distance x 
from the axis, and let y be the vertical 
distance of G above the lowest point C. 

The head of water on any point E in the 
base is EG or hi + y. Now this head y 
is caused by the velocity u with which 
the point G revolves around the axis, or, in other words, the 
position of G above C is due to the energy of rotation. Thus if 
W is the weight of a particle of water at G, the potential energy 
IVy equals the kinetic energy Wu 2 /2g, and hence y = u 2 /2g. 

Let n be the number of revolutions made by the vessel and 
water in one second. Then u = 21rx • n, and hence 

y = u 2 / 2g — 2 7 r 2 n 2 x 2 /g 

which is the equation of a common parabola with respect to rec¬ 
tangular axes having an origin at its vertex C. The surface of 
revolution is hence a paraboloid. 

Since the volume of a paraboloid is one-half that of its circum¬ 
scribing cylinder, and since the same quantity of water is in the 
vessel when in motion as when at rest, it is plain that in the 
figure \{h 2 — /q) equals h — /q. Consequently h — /q equals 
/q — h , or the elevation of the water surface at D above its original 
level is equal to its depression at C. If r be the radius of the ves¬ 
sel, the height /q — hi is, from the above equation, 2 7 r 2 n 2 r 2 /g, and 
hence the distances h — hi and h 2 — h are each equal to ir 2 n 2 r 2 /g. 
The head at the middle of the base of the vessel during the motion 
is now hi = h — ir 2 n 2 r 2 /g and the head at any point E is hi + y = 
h + (2X 2 — r 2 )ir 2 n 2 !g. 

The theoretic velocity of flow from the small orifice in the 
base is that due to the head /q + y, or 

V = V2g (hi + y) = V 2 gh + 27 T 2 n 2 ( 2 x 2 — r 2 ) 



Fig. 29a. 









04 


Chap. 3. Theoretical Hydraulics 


which is less than Vg gh when x 2 is less than \r\ and greater 
when is greater than \ r 2 . For example, let r = 1 foot and 
h = 3 feet, then V = 13.9 feet per second when the vessel is at 
rest. But if it is rotating three times per second around its axis 
with uniform speed, the velocity from an orifice in the center of 
the base, where x = o, is 3.9 feet per second, while the velocity 
from an orifice at the circumference of the base, where x = 1 foot, 
is 19.2 feet per second. At this speed the water is depressed 2.76 
feet below its original level at the center and elevated the same 
amount above that level around the sides of the vessel. 

In the case of a closed vessel where the paraboloid cannot form, 
the velocity of flow from all orifices, except one at the axis, is 
increased by the rotation. Thus in Fig. 29 b, if 
the vessel is at rest and the head on the base is h , 
the velocity of flow from all small orifices in the 
base is V2 gh. But if the vessel is revolved about 
the vertical axis BC, so that an orifice at E has the 
velocity u around that axis, then the pressure-head at E is 
h + u 2 / 2g, and accordingly 

V = V 2 gh + u 2 ( 29 ) 

is the theoretic velocity of flow from an orifice at E. This formula 
is an important one in the discussion of hydraulic motors. Here, 
as before, the value of u may be expressed as 27 rxn, when x is 
the distance of E from the axis and n is the number of revolutions 
per second. As an example, let a closed vessel full of water be 
revolved about an axis 120 times per minute, and let it be re¬ 
quired to find the theoretic velocity of flow from an orifice ij 
feet from the axis, the head on which is 4 feet when the vessel is at 
rest. The velocity u is found to be 18.85 f ee t P er second, and 
then the theoretic velocity of flow from the orifice is 24.8 feet per 
second, whereas it is only 16 feet per second when the vessel 
is at rest. 

The velocity V in both these cases is a relative velocity, for the 
pressure at the moving orifice produces a velocity with respect to 
the vessel. The absolute velocity, or that with respect to the 
earth, is greater than the relative velocity when the stream issues 



Fig. 29 b . 
















Theoretic Discharge. Art. 30 


65 


from an orifice in the base, for the orifice moves horizontally with 
the absolute velocity u and the stream moves downward with the 
relat ive veloc ity V, and hence the absolute velocity of the stream 
is V V 2 + u 2 . When the stream issues from an orifice in the side 
of the vessel upon which the head is h, formula ( 29 ) gives its rela¬ 
tive velocity, and then the absolute velocity is found by ( 28 ). 

Prob. 29 . A cylindrical vessel 2 feet in diameter and 3 feet deep is three- 
fourths full of water, and is revolved about its vertical axis so that the water 
is just on the point of overflowing around the upper edge. Find the number 
of revolutions per minute. Find the relative velocity of flow from an orifice 
in the base at a distance of 0.75 foot from the axis. Show that the velocity 
from all orifices within 0.707 foot of the axis is less than if the vessel were 
at rest. 


Art. 30 . Theoretic Discharge 

The term ‘ ‘ discharge ” means the volume of water flowing 
in one second from a pipe or orifice, and the letter Q will designate 
the theoretic discharge; that is, the discharge as computed with¬ 
out considering the losses due to frictional resistances. When 
all the filaments of water issue from the pipe or orifice with the 
same velocity, the quantity of water issuing in one second is 
equal to the volume of a prism having a base equal to the cross- 
section of the stream and a length equal to the velocity. If 
this area is a and the theoretic velocity is V, then Q = aV is 
the theoretic discharge. Taking a in square feet and V in feet 
per second, the discharge Q is in cubic feet per second. 

For a small orifice on which the head h has the same value 
for all parts of the opening, the theoretic discharge is 

Q = aV = a y/2gh ( 30 ) 

and in English measures Q = 8.02 a ~\fh. For example, let a 
circular orifice 3 inches in diameter be under a head of 10.5 feet, 
and let it be required to compute Q. Here 3 inches = 0.25 foot 
and from Table F the area of the circle is 0.04909 square foot. 
From Art. 22 the theoretic velocity V is 8.02 X V10.5 = 25.99 
feet per second. Accordingly the theoretic discharge is 0.04909 
X 25.99 = !- 2 8 cubic feet per second. 




66 


Chap. 3. Theoretical Hydraulics 


The above formula for Q applies strictly only to horizontal 
orifices upon which the head h is constant, but it will be seen 
later that its error for vertical orifices is less than one-half of one 
percent when h is greater than double the depth of the orifice. 
Horizontal orifices are but little used, as it is more convenient 
in practice to arrange an opening in the side of a vessel than in 
its base. In applying the above formula to a vertical orifice, h is 
taken as the vertical distance from its center to the free-water 
surface. Vertical orifices where the head h is small are discussed 
in Arts. 47 and 48 . 

Since the theoretic velocity is always greater than the actual 
velocity, the theoretic discharge is a limit which can never be 
reached under actual conditions. Theoretically the discharge 
is independent of the shape of the orifice, so that a square orifice 
of area a gives the same theoretic discharge as a circular orifice 
of area a ; it will be seen in Chap. 5 that this is not quite true for 
the actual discharge. 

In this chapter it is supposed that the velocity of a jet is the 
same in all parts of the cross-section, as this would be the case if 
h has the same value throughout the section were it not for the 
retarding influence of friction. Actually, however, the filaments 
of water near the edges of the orifice move slower than those 
near the center. If q be the actual discharge from any orifice and 
v the mean velocity in the area a , then q = av, or the equation 
v = q/a may be regarded as a definition of the term “mean 
velocity. ” The theoretic mean velocity is 2 ~Vgh, but the actual 
mean velocity is slightly smaller, as will be seen in Chap. 5 . 

Formula ( 30 ) may be used for computing h when Q and a 
are given, and it shows that the theoretic head required to de¬ 
liver a given discharge varies inversely as the square of the area 
of the orifice. 

Prob. 30 a. Compute the theoretic head required to deliver 300 
gallons of water per minute through an orifice 3 inches in diameter. 

Prob. 30 b. A vessel one foot square has a small orifice in the base. 
What is the theoretic velocity of flow from this orifice when the vessel con¬ 
tains 125 pounds of mercury? Also when it contains 250 pounds of water? 


Steady Flow in Smooth Pipes. Art. 31 


G 7 


Art. 31 . Steady Flow in Smooth Pipes 

When water flows through a pipe of varying cross-section 
and all sections are filled with water, the same quantity of water 
passes each section in one second. This is called the case of steady 
flow. Let q be this quantity of water and let Vi, V2, Vs be the mean 
velocities in three sections whose areas are a\, a 2 , as. Then 

q = aiv 1 = a 2 v 2 = a 3 v 3 (31)i 

This is called the condition for steady flow or the equation of 
continuity, and it shows that the velocities at different sections 
vary inversely as the areas of those sections. If v be the velocity 
at the end of the pipe where the area is a, then also q = av. 
When the discharge q and the areas of the cross-sections have 
been measured, the mean velocities may be computed. 

When a pipe is filled with water at rest, the pressure at any 
point depends only upon the head of water above that point. 
But when the water is in motion, it is a fact of observation that 
the pressure becomes less than that due to the head. The unit- 
pressure in any case may be measured by the height of a column 
of water. Thus if water be 
at rest in the case shown in 
Fig. 31 a, and small tubes be 
inserted at the sections whose 
areas are a\ and a 2 , the water 
will rise in each tube to the 
same level as that of the water 
surface in the reservoir, and 
the pressures in the sections 
will be those due to the hydrostatic heads Hi and H 2 . But it 
the valve at the right be opened, the water levels in the small 
tubes will sink and the mean pressures in the two sections will be 
those due to the pressure-heads h and h 2 . 

Let W be the weight of water flowing in each second through 
each section of the pipe, and let vi and v 2 be the mean velocity 
in the section ai and a 2 . When this water was at rest, the poten¬ 
tial energy of pressure in the section a< was WH \; when it is in 

















68 


Chap. 3. Theoretical Hydraulics 


motion, the energy in the section is the pressure energy Wh\ plus 
the kinetic energy W • v x /2g. If no losses of energy clue to fric¬ 
tion or impact have occurred, the energy in the two cases must be 
equal. The same reasoning applies to the section a 2 , and hence 

Hi = //i + — and H i = h + — ( 31 )* 

2g 2 g 

These equations exhibit the law of steady flow first deduced by 
Daniel Bernouilli in 1738, and hence often called Bernouilli’s 
theorem ; it may be stated in words as follows : 

At any section of a tube or pipe, under steady flow without 
friction, the pressure-head plus the velocity-head is equal to 
the hydrostatic head that obtains when there is no flow. 

This theorem of theoretical hydraulics is of great importance in 
practice, although it has been deduced for mean velocities and 
mean pressure-heads, while actually the velocity and the pressure 
are not the same for all points of the cross-section. 

The pressure-head at any section hence decreases when the 
velocity of the water increases. To illustrate, let the depths 
of the centers of a Y and a 2 be 6 and 8 feet below the water level, 
and let their areas be 1.2 and 2.4 square feet. Let the discharge 
of the pipe be 14.4 cubic feet per second. Then from ( 31 ) 1 the 
mean velocity in a x is v\ = 14.4/1.2 = 12 feet per second, which 
corresponds to a velocity head of 0.01555Z; 2 = 2.24 feet, and 
consequently from (31 ) 2 the pressure-head in a x is 6.0 — 2.24 = 
3.76 feet. For the section a 2 the velocity is 6 feet per second and 
the velocity head is 0.56 feet, so that the pressure-head there is 
8.0 — 0.56 = 7.44 feet. 

The theorem of ( 31)2 may be also applied to the jet issuing 
from the end of the pipe. Outside the pipe there can be no pres¬ 
sure, and if h be the hydrostatic head and V the velocity, the 
equation gives li = V 2 /2g , or V = V2 gh ; that is, if frictional 
resistances be not considered, the theoretic velocity of flow from 
the end of a pipe is that due to the hydrostatic head upon it. In 
Chap. 8 it will be seen that the actual velocity is much smaller 



Emptying a Vessel. Art. 32 


GO 


than this, lor a large part of the head h is expended in over¬ 
coming friction in the pipe. 

A negative pressure may occur if the velocity-head becomes 
greater than the hydrostatic head, for ( 31) 2 shows that //, is 
negative when vf/2g exceeds H Y . A case of this kind is given 
in Fig. 316, where the section at A is so small that the velocitv 
is greater than that due to the head H lf 
so that if a tube be inserted at A, no water 
runs out; but if the tube be carried down¬ 
ward into a vessel of water, there will be 
lifted a column CD whose height is that 
of the negative pressure-head h\. For ex¬ 
ample, let the cross-section of A be 0.4 
square feet, and its head H 1 be 4.1 feet, while 8 cubic feet 
per second are discharged from the orifice below. Then the 
velocity at A is 20 feet per second, and the corresponding ve¬ 
locity-head is 6.22 feet. The pressure head at A then is, from 
the theorem of formula ( 31 ) 2 , 

h\ = 4.1 — 6.22 = — 2.12 feet 

and accordingly there exists at A an inward pressure 

pi — — 2.12 X 0.434 = — 0.92 pounds per square inch 

This negative pressure will sustain a column of water CD whose 
height is 2.12 feet. When the small vessel is placed so that its 
water level is less than 2.12 feet below A, water will be constantly 
drawn from the smaller to the larger vessel. This is the principle 
of the action of the injector-pump. 

Prob. 31 . In a horizontal tube there are two sections of diameters 1.0 
and 1.5 feet. The velocity in the first section is 6.32 feet per second, and 
the pressure-head is 21.57 feet. Find the pressure-head for the second sec¬ 
tion if no energy is lost between the sections. 

Art. 32. Emptying a Vessel 

Let the depth of water in a vessel be 11 ; it is required to de¬ 
termine the theoretic time of emptying it through an orifice in 
the base whose area is a. Let Y be the area of the water surface 














70 


Chap. 3. Theoretical Hydraulics 


when the depth of water is y; let St be the time during which 
the water level falls the distance 8 y. During this time the quan¬ 
tity of water Y • 8 y passes through the orifice. But the dis¬ 
charge in one second under the constant head y is a V2gy, and 
hence the discharge in the time St is aSt^/2gy. Equating these 
two expressions, there is found the general formula 
which gives the time for the water surface to drop 

the distance 8 y } yg 

St = -*— 



a 


V2 gy 


(32), 


Fig. 32 a . 


The time of emptying any vessel is now deter¬ 
mined by inserting for Y its value in terms of y, and then in¬ 
tegrating between the limits H and o. 


For a cylinder or prism the cross-section Y has the constant 
value A , and the formula becomes 

u = 4ylh 

a V 2g 

the integration of which, between limits H and h, gives 


d\ 2g 

as the theoretic time for the head H to fall to h. If h = o, this 
formula gives the time of emptying the vessel. If the head were 
maintained constant, the uniform discharge per second would 
be a V2 gH, and the time of discharging a quantity equal to the 
capacity of the vessel is AH divided by a^/2gH, which is one- 
half of the time required to empty it. 


To find the time of emptying a hemispherical bowl of radius 
r through a small orifice at its lowest point, let x be the radius 
of the cross-section Y ; then x 2 + (r — y) 2 = r 2 is the equation 
of the circle, from which the area F is 7r(2ry — y 2 ). Then 

St = ——— (2 ry 1 — y 1 ) Sy 
a\2g 

and by integration between the limits r and o 

t = 147 rrV 15a V2 ~g 

which is the theoretic time required to empty the bowl. 


















Emptying a Vessel. Art. 32 


71 


The most important application of these principles is in the 
case of the right prism or cylinder, and here the formula for the 
time is modified in practice by introducing a coefficient, as may 
be seen in Art. 58 . The theoretic time found by the above for¬ 
mula is always too small, since frictional resistances have not been 
considered. Moreover, the formula does not strictly apply when 
the head is very small, owing to a whirling motion that occurs and 
which tends to increase the theoretic time. 

Venturi, in 1798, first described the phenomena of this whirl.* 
When the head becomes less than about three diameters of the orifice, 
the water is observed in whirling motion, the velocity being greatest 
near the vertical axis through the center of the orifice, and as the head 
decreases a funnel is formed through the middle of the issuing stream. 
The direction of this w T hirl, as seen from above, may be either clock¬ 
wise or contraclockwise, depending on initial motions in the w T ater 
or on irregularities in the vessel or orifice, but under ideal conditions 
it should be clockwise in the southern hemisphere of the earth and con¬ 
traclockwise in the northern hemisphere, this being the effect of 
the earth’s rotation. Fig. 326 repre¬ 
sents a vertical section of this funnel, 
on which A is any point having the 
coordinates x and y with respect to 
the rectangular axes OX and OY. The 
axis OY is drawn through the center of 
the orifice, and OX is tangent to the 
level water surface at a distance H 
above the bottom of the vessel. Let r 
be the radius of the funnel in the plane of the orifice. It is required 
to find the relation between x, y, H, and r, or the equation of the 
curve shown in the figure. 

An approximate solution may be made by supposing that the par¬ 
ticle of water at A is moving nearly horizontally around the axis 
OF with the velocity v ; this velocity must be due to the head y, 
whence v 2 = 2gy. This particle is acted upon by the downward force 
AB, due to gravity, and by the horizontal force AC, due to centrif¬ 
ugal action, and they are proportional to g and v 2 /x, these being the 

* Tredgold’s Tracts on Hydraulics (London, 1799 and 1826 ) gives a 

translation of the memoir of Venturi. 












72 


Chap. 3. Theoretical Hydraulics 


accelerations due to gravity and centrifugal force. The ratio AC/AB 
is the tangent of the angle # which the water surface at A makes with 
the axis OX, for this surface must be normal to the resultant AD of the 
two forces AB and AC. When the ordinate y is increased to y + 8y, 
the abscissa x is decreased to x—Sx, and hence the value of tan# 
must be the same as —Sy/Sx. Accordingly 


tan# 


AC = f_ 
A B gx 



8v 


and the integration of this differential equation gives y = C/x 2 , in 
which C is the constant of integration. When y equals H . the value 
of x is r, and hence C = Hr 2 , and thus 


y = Hr 2 /x 2 


( 32) 2 


is the equation of the curve, which may be called a quadratic hyper¬ 
bola, the surface of the funnel being then a quadratic hyperboloid. 
This equation represents the curve at one instant only, for II contin¬ 
ually decreases as the water flows out, since the direction of v is not 
quite horizontal as the investigation assumes. The general phenom¬ 
ena are, however, well explained by this discussion. 

Prob. 32 . A prismatic vessel has a cross-section of 18 square feet and 
an orifice in its base has an area of 0.18 square foot. Find the theoretic 
time for the water level to drop 7 feet, when the head upon the orifice at the 
beginning is 16 feet. 


Art. 33. Computations in Metric Measures 

(Art. 22 ) Using for the acceleration of the mean value 9.80 
meters per second per second, formulas (22) 2 become 

V = 4.427 V/z h = 0.05102 V 2 ( 33 ) 

in which h is in meters and V in meters per second. Table 33 shows 
values of the velocity for given heads, and values of the velocity-head 
for given velocities. 

(Art. 23 ) For Fig. 23 let the reservoir be one meter in diameter, 
the load W be 2000 kilograms, and the orifices be 3 meters below the 
piston. Let the exterior head on A be 1.5 meters, the orifice B be 
open to the atmosphere, and the orifice C be in air whose pressure is 
0.7 kilograms per square centimeter. The area of the piston is 0.7854 


Computations in Metric Measures. Art. 33 


73 


Table 33. Velocities and Velocity-heads 


Metric Measures 



V =V2 gh 

= 4427 Va 


II 

c* 

II 

* 5 ; 

0.05102 V 2 


Head in 
Meters 

Velocity 
in Meters 
per Second 

Head in 
Meters 

Velocity 
in Meters 
per Second 

Velocity 

in 

Meters 

per 

Second 

Head in 
Meters 

Velocity 

in 

Meters 

per 

Second 

Head in 
Meters 

0.1 

1-432 

1 

4-427 

0.1 

0.0005 

1 

0.0510 

0.2 

1.980 

2 

6.262 

0.2 

0.0020 

2 

O.2041 

O.J 

2.425 

3 

7.668 

0-3 

0.0046 

3 

O.4592 

0.4 

2-799 

4 

8.854 

0.4 

0.0082 

4 

0.8163 

0.5 

3 -I 3 I 

5 

9.900 

0-5 

0.0123 

5 

1.276 

0.6 

3-429 

6 

IO.84 

0.6 

0.0184 

6 

1-837 

0.7 

3-704 

7 

II.71 

0.7 

0.0250 

7 

2.500 

0.8 

3.960 

8 

12.52 

0.8 

0.0327 

8 

3-265 

0-9 

4.200 

9 

13.28 

0.9 

0.0413 

9 

4-133 

1.0 

4-427 

10 

14.OO 

1.0 

°-° 5 10 

10 

5.102 


square meters, and the head corresponding to the pressure on the upper 
water surface is 



w 


2000 

0.7854 X 1000 


= 2.546 meters. 


The head /q is 3 meters for the first orifice, o for the second, and —10 
(1.033—0.7) = — 3.33 meters for the third. The three theoretic 
velocities of outflow then are 

V = 4.427 V3 + 2.546 — 1.5 = 8.91 meters per second, 

V = 4.427 A/3 + 2.546 — o =10.43 meters per second, 

V = 4.427 V3 + .546 + 3.33 = 13.19 meters per second. 


If in this example the liquid be alcohol which weighs 800 kilograms per 
cubic meter, the head of alcohol corresponding to the pressure of the 

P iston is . onnn 

= 3.183 meters, 


h = 


0.7854 X 800 


and accordingly for discharge into the atmosphere at the depth 
hi = 3 meters the velocity is 

V = 4.427 V3 +3.18 = 1 r.01 meters per second, 


while for water the velocity was 10.43 meters per second. 




































74 


Chap. 3. Theoretical Hydraulics 


(Art. 26 ) As an illustration of ( 26) 2 let water issue from a pipe 
6 centimeters in diameter with a velocity of 4 meters per second. The 
cross-section is found from Table F to be 0.002827 square meters, 
and then the theoretic work in kilogram-meters per second is 

K = 0.05102 X 1000 X 0.002827 X 4 3 = 9.23 

which is 0.123 metric horse-power. If the velocity is 16 meters 
per second, the stream will furnish 7.87 horse-powers. 

(Art. 30 ) The area a is in square meters, the velocity V in meters 
per second, and the discharge Q in cubic meters per second. Thus 
if a pipe 20 centimeters in diameter discharges 0.15 cubic meters per 
second, the area of the cross-section is 0.03142 square meters and the 
mean velocity is 0.15/0.03142 = 4.77 meters per second. 

(Art. 31 ) In Fig. 31 a, suppose the sections a x and a 2 to be 0.06 
and 0.12 square meters, and the depths of their centers below the 
water level of the reservoir to be 4.5 and 5.5 meters. Let 0.24 cubic 
meters per second be discharged from the pipe, then from ( 31 ) x 
the mean velocities in a x and a 2 are 4.0 and 2.0 meters per second. 
The velocity-heads are then 0.82 meters for ai and 0.20 meters for a 2 , 
so that during the flow the pressure-head at A is 4.5 — 0.82 = 3.68 
meters and that at B is 5.5 — 0.20 = 5.30 meters. 

Prob. 33 a. What theoretic velocities are produced by heads of 0.1, 
0.01, and 0.001 meter? What is the velocity-head of a jet, 7.5 centimeters 
in diameter, which discharges 500 liters per second ? 

Prob. 33 b. A prismatic vessel has a cross-section of 1.5 square meters 
and an orifice in its base has an area of 150 square centimeters. Compute 
the theoretic time for the water level to drop 3 meters when the head at the 
beginning is 4 meters. 

Prob. 33 c. A small turbine wheel using 3 cubic meters of water per 
minute under a head of io£ meters is found to deliver 5.1 metric horse¬ 
powers. Compute the efficiency of the wheel. 

Prob. 33 d. In an inclined tube there are two sections of diameters 10 
and 20 centimeters, the second section being 1.536 meters higher than the 
first. The velocity in the first section is 6 meters per second and the pres¬ 
sure-head is 7.045 meters. Find the pressure-head for the second section. 


General Considerations. Art. 34 


,75 


CHAPTER 4 


INSTRUMENTS AND OBSERVATIONS 
Art. 34 . General Considerations 


Some of the most important practical problems of Hydraulics 
are those involving the measurement of the amount of water dis¬ 
charged in one second from an orifice, pipe, or conduit under given 
conditions. The theoretic formulas of the last chapter furnish 
the basis of most of these methods, and in the chapters following 
this one are given coefficients derived from experience which 
enable those formulas to be applied to practical conditions. 
These coefficients have been determined by measuring heads, 
pressures, or velocities with certain instruments, and also the 
amount of water actually discharged, and then comparing the 
theoretic results with the actual ones. It is the main object of 
this chapter to describe the instruments used for this purpose, and 
a few remarks concerning advantageous methods for the discus¬ 
sion of the observations will also be made. 



My 


The engineer’s steel tape, level, and transit are indispensable 
tools in many practical hydraulic problems. For example, two 
reservoirs M and N, connected by a pipe line, may be several 
miles apart. To ascertain the difference in elevation of their 
water surfaces lines of levels may 
be run and bench marks established 
near each reservoir, as also at other 
points along the pipe line. From 
the bench marks at the reservoirs 
there can be set up simple board 
gages, so that simultaneous read¬ 
ings can be taken at any time to find the difference in eleva¬ 
tion. From the bench marks along the pipe line a profile of the 
same can be plotted for use in the discussion. With the transit 



Fig. 34a. 









































7 G 


Chap. 4. Instruments and Observations 


and tape the alignment of the pipe line and the lengths of its 
curves and tangents can also be taken and mapped. All of 
these records, in fact, are necessary in order to determine the 
amount of water delivered through the pipe. 

For work on a smaller scale, like that of the discharge from 
an orifice in a tank, the steel tape may be used to mark points 

from which a glass gage tube may be set and 
upon which the height of the water surface 
above the orifice can be read at any time during 
the experiment. Another method is to have a 
float on the water surface, the vertical motion 
of which is communicated to a cord passing over 
a pulley, so that readings can be taken on a scale 
as the weight at the lower end of the cord 
moves up or down. When the head is very small, however, 
these methods are not sufficiently precise, and the hook gage 
described in Art. 35 must be used. 



It is often desirable for many purposes to keep a continuous 
record of the level of a water surface. This can be accom¬ 



plished by the use of an automatic re¬ 
cording gage such as that shown in 
Fig. 34 c. This apparatus, as made by 
Freiz, consists essentially of a float con¬ 
nected to a flat perforated copper band 
which passes over a sprocket wheel and 


Fig. 34c. 











































































































































General Considerations. Art. 34 77 

which carries at its other end a counterweight. The sprocket 
wheel is directly connected to a drum the circumference of which 
is exactly one foot and on which a sheet of ruled paper can be 
clamped. A clockwork moves a pen at a constant and uniform 
rate in a direction parallel to the axis of the cylinder, and if the 
latter remains stationary, the pen will draw a straight line on 
the paper. If, however, the cylinder is caused to revolve by the 
rising or falling of the float, the pen will draw a curve, and each 
revolution of the cylinder will represent a change of one foot in 
the water level. Each sheet or chart, depending on the gear 
of the clock, will give a record either 24 hours or 7 days long 
before a new chart must be put on by an attendant. By the 
interposition of suitable gears between the sprocket wheel and 
the cylinder the ratio of the number of revolutions between 
the sprocket and the drum can be fixed at any desired number. 
With all forms of apparatus of this kind it is desirable that the 
float should be of large horizontal diameter in order that its lift¬ 
ing power may be sufficient to overcome the friction in the bear¬ 
ings of the machine and so cause it to easily and quickly 
respond to small fluctuations in the water surface. 

The Bristol recording water level gage operates on the principle 
of the aneroid barometer. A bronze cylindrical box encloses air, the 
pressure of which is communicated through a flexible tube to the re¬ 
cording apparatus whenever that pressure exceeds the exterior atmos¬ 
pheric pressure. When this box is placed under water, the head of 
water acts on a diaphragm and increases the air pressure an amount 
proportional to the head on the diaphragm. In the recording ap¬ 
paratus is a pen which draws a curve on a sheet of paper moved by * 
clockwork and thus gives a continuous record of the water level. 
This apparatus has been used for recording the heights of tides and 
of water levels in reservoirs. Of course the adjustment of the 
instrument must be made by experiment, its record being compared 
by one made by direct methods. The closest reliable reading of 
a gage of this kind appears to be about one-eighth of an inch. 

A small quantity of water flowing from an orifice may be 
measured by allowing it to run into a barrel set upon a platform 
weighing scale. The weight of water discharged in a given time 


78 


Chap. 4. Instruments and Observations 


is thus ascertained, the time being noted by a stop-watch, and 
the volume is then computed by the help of Table 3 . If the flow 
is uniform, the discharge in one second is then found by dividing 
the volume by the number of seconds. A larger quantity of 
water may be measured in a rectangular tank, the cross-section 
of which is accurately known; here the water surface is noted at 
the beginning and end of the experiment, and the volume is then 
computed by multiplying the area by the differences of the two 
elevations. For example, a square tank was 4 feet 2 inches in¬ 
side dimensions, and the gage read 3.17 feet at the beginning and 
4.62 feet at the end of the experiment, which lasted 304 sec¬ 
onds ; then the flow, if uniform, was 0.0828 cubic feet per second. 

Larger quantities of water still are sometimes measured in 
the reservoir of a city supply. The engineer, by the use of his 
level, transit, and tape, makes a precise contour map of the 
reservoir, determines with the planimeter the area enclosed by 

each contour curve, and com¬ 
putes the volume included 
between successive contour 
planes. For instance, if the 
area of the contour curve A B 
is 84 320 square feet and that 
of CD is 79 624 square feet and 
the vertical distance between 
the contour planes is 5 feet, 
the volume included is 409 S60 
cubic feet by the method of 
mean areas. A more precise determination, however, may be 
made by measuring the area of a contour curve halfway between 
AB and AC; if this is found to be 82 150 square feet, the volume 
included between AB and AC is computed by the prismoidal 
formula and found to be 410 450 cubic feet. 

These direct methods of water measurement form the basis of 
all hydraulic practice. In this manner water meters are rated, and 
the coefficients determined by which practical formulas for flow through 
orifices, weirs, and pipes are established. These coefficients being 
known, indirect methods may be used for water measurement; namely, 


















The Hook Gage. Art. 35 


79 


the discharge can be computed from the formulas after area and heads 
have been ascertained. There are also methods of indirect measure¬ 
ment from observed velocities which will be described later, and which 
are especially valuable in finding the discharge of conduits and streams. 

Prob. 34 . Water flows from an orifice uniformly for 89.3 seconds and falls 
into a barrel on a platform weighing scale. The weight of the empty barrel 
is 27 pounds and that of the barrel and water is 276 pounds. What is the 
discharge of the orifice in gallons per minute, when the temperature of the 
water is 62° Fahrenheit ? 


Art. 35 . The Hook Gage 


H8 


The hook gage, invented by Boyden about 1840, consists of 
a graduated metallic rod sliding vertically in fixed supports, upon 
which is a vernier by which readings can be taken to 
thousandths of a foot. At the lower end of the rod is 
a sharp-pointed hook, which is raised or lowered until 
its point is at the water level. Fig. 35 a represents 
the form of hook gage made by Gurley, the gradua¬ 
tion on the rod being to feet and hundredths. The 
graduation has a length of 2.2 feet, so that variations in 
the water level of less than this amount can be meas¬ 
ured, by using the vernier, to thousandths of a foot. To 
take a reading on a water surface, the point of the hook 
is lowered below the surface and then slowly raised by 
the screw at the top of the instrument. Just before 
the point of the hook pierces the skin of the water 
(Art. 2 ) a pimple or protuberance is seen to rise above 
it; the hook is then depressed until the pimple is 
barely visible and the vernier is read. The most pre¬ 
cise hook gages read to ten-thousandths of a foot, and it 
has been stated that an experienced observer can, in a 
favorable light and on a water surface perfectly quiet, 
detect differences of level as small as 0.0002 feet. 


Fig. 35a. 


A cheaper form of hook gage, and one sufficiently pre¬ 
cise for many classes of work, can be made by screwing a 
hook into the foot of an engineer’s leveling rod. The back part of 
the rod is then held in a vertical position by two clamps on fixed 

































80 


Chap. 4. Instruments and Observations 


supports, while the front part is free to slide. It is easy to arrange 
a slow-motion movement so that the point of the hook may be 
precisely placed at the water level. The reading of the vernier is 
determined when the point of the hook is at a known elevation 
above an orifice or the crest of a weir, and by subtracting from this 
the subsequent readings the heads of water are known. A New' 
York leveling rod, reading to thousandths of a foot on its vernier, 
is to be preferred for this work. 

Hook gages are principally used for determining the eleva¬ 
tions of thC water surface above the crest of a weir, as the heads 
of water are small and must be known with precision. In Fig. 
35 b, the crest of the weir is seen and the hook gage is erected at 

some distance back from it, where the 
water surface is level. In this case great 
care should be taken to determine the read¬ 
ing corresponding to the level of the crest. 
In the larger forms of hooks this may be 
done by taking elevations of the crest and 
of the point of the hook by means of an 
engineer’s level and a light rod. With smaller hooks it may be 
done by having a stiff permanent hook, the elevation of whose 
point with respect to the crest is determined by precise levels; 
the water is then allowed to rise slowly until it reaches the 
point of this stiff hook, when readings of the vernier of the 
lighter hook are taken. Another method is to allow a small 
depth of water to flow over the crest and to take readings of the 
hook, while at the same time the depth on the crest is measured 
by a finely graduated scale. Still another way is to allow the 
water to rise slowly, and to set the hook at the water level when 
the first filaments pass over the crest; this method is not a very 
precise one on account of capillary attraction along the crest. 
As the error in setting the hook is a constant one which affects 
all the subsequent observations, especial care should be taken to 
reduce it to a minimum by taking a number of observations in 
order to obtain a precise mean result. 

The hook gage is also used to find the difference oi the water 
levels in tanks for experiments for the determination of hydraulic 










Pressure Gages. Art. 36 


81 


coefficients, and in wells along pipe lines when experiments are 
made to investigate frictional resistances. In general its use is 
confined to cases where the head is small, as for high heads so 
great a degree of precision is not required (Art. 54 ). 

Prob. 35 . A wooden tank, 4.52 by 5.78 feet in inside dimensions, has 
leakage near its base. The hook gage reads 2.047 f ee t at 11.57 a.m., 
1.470 feet at 12.05 P - M -, and 0.938 foot at 12.13 p .M. Compute the probable 
leakage in the first and last minutes. 


Art. 36. Pressure Gages 

A pressure gage, often called a piezometer, is an instru¬ 
ment for measuring the pressure of water in a pipe. The form 
most commonly found in the market has a dial and movable 
pointer, the dial being graduated to read pounds per square inch. 
The principle on which this gage acts is the same as that of the 
Richard aneroid barometer and the Bourdon steam gage. 
Within the case is a small coiled tube closed at one end. while the 
other end is attached to the opening through which the water is 
admitted. This tube has a tendency to straighten when under 
pressure, and thus its closed end moves and the motion is com¬ 
municated to the pointer; when the pressure is relieved, the tube 
assumes its original position and the pointer returns to zero. 
There is no theoretical method of determining the motion of the 
pointer due to a given pressure, and this is done by tests in which 
known pressures are employed, and accordingly the divisions 
on the graduated scale are usually unequal. These gages are 
liable to error after having been in use for some time, especially 
so at high pressures, and hence should be tested before and after 
any important series of experiments. 

In most hydraulic work the head of water causing the pressure 
is required to be known. When p is the gage reading in pounds 
per square inch, the head of water in feet is h = 2.304 p, or when 
p is the gage reading in kilograms per square centimeter, the head 
of water in meters is h = to p. The graduation of the gage dial 
may be made to read heads directly, so as to avoid the necessity 
of numerical reduction. 


82 


Chap. 4. Instruments and Observations 



Fig. 36a. 


The pressure at any point of a pipe may be measured by the 
height of a column of water in an open tube, as seen at A in Fig. 
36 a. The upper portion of the tube may be of glass, so that the 

position of the water level may be 
noted on a scale held alongside. 
It is not necessary that the water 
column should be vertical, and a 
hose is often used, as seen at B , 
with a glass tube at its top. At 
C is shown a dial pressure gage. 
When the head h is directly read 
in feet, the pressure in pounds 
per square inch may be computed from p = 0.434//. In order 
to secure precise results when the water in the pipe is in motion, 
it is necessary that a piezometer tube be inserted into the pipe 
at right angles; when inclined toward or against the current, 
the head h is greater or less than that due to the actual pressure 
at its mouth. 

For high pressures a water column is impracticable on ac¬ 
count of its great height, and hence mercury gages are used. 
Fig. 366 shows the principle of construction, a bent tube ABC 
with both ends open, having mercury in its lower 
portion, and the water column of height h being 
balanced by the mercury column of height z. If 
the atmospheric pressures at A and C are the same, 
it is evident, from Art. 4 , that the height h is about 
13.6 times the height z, since the specific gravity 
of mercury is about 13.6. Now 2 can be read on 
a scale placed between the legs of the tube, and 
thus h is known, as also the water pressure at the 
point B. If the atmospheric pressures at A and C 
are different, as will be the case when h is very large, 
let b x be the barometer reading at A and fa that at 
C, both being in the same linear unit as h and z. 

The absolute pressure at B is that due to the 
height sh + s'b x , where s and s' are the specific 
gravities of water and mercury, and the absolute 


A. |T- ■ 


h 


B 


if 

1 z 


W 

Fig. 366. 



















































Pressure Gages. Art. 36 


83 


pressure at the same elevation in the other leg is that due to 
the height s'(z + 62). Since these pressures are equal, 

h = (s' /s)(z + b 2 — bi) 

is the head corresponding to the distance z on the scale. The 
ratio s'/s is 13.6 approximately, its actual value depending on the 
purity of the water and mercury and on the temperature. 

Fig. 36 c shows the mercury gage as arranged for measuring 
the pressure-head at a point A in a water pipe. The top is open 
to the air and through it the mercury may be poured in, the cock 
E being closed and F open ; the mercury then stands at the same 
height in each tube. The cock F being closed and E opened, 
the water enters the left-hand tube, depressing the mercury to 




B , causing it to rise to C on the other side. The distance z is 
then read on a scale between the two tubes, and the height of 
B above A by another scale. The pressure of the water at B 
is that due to the head 13.62, and the pressure at A is that due to 
the head y -f 13.6s. In precise work it is necessary to deter¬ 
mine the exact specific gravity of the mercury and water at dif¬ 
ferent temperatures, so that precise values of the ratio s'/s may 
be known. The value of 5' depends upon the purity of the mer¬ 
cury and is sometimes lower than 13.56. 

A better form of mercury gage for use under most conditions 
is shown in Fig. 3 6 d. It consists essentially of a heavy cast-iron 
reservoir having 3 large horizontal cross-section as compared with 







































84 Chap. 4 . Instruments and Observations 

that of the glass tube T. The surface of the mercury M in this 

reservoir therefore remains at a practically constant level, and 

this level can be seen through a small glass window provided 
* 

for that purpose. The glass tube is inserted through a stuffing 
box at S and the flow of mercury into it is controlled by a valve 
at C. Cocks at A permit of drawing off and preventing the 
entrainment of air, and the water pressure is admitted to the gage 
through the valve B. In case observations are to be made on 
a pressure which is constantly fluctuating the resulting oscilla¬ 
tions in the tube can be dampened by partially closing the valves 
at either or both B and C. 

For very high pressures, such as are used in operating heavy 
forging-presses, the mercury column of the above gage would be so 
long as to render it impracticable, and accordingly other methods must 
be employed. Fig. 36c represents a mercury gage constructed on 

the principle of the hydraulic press 
(Art. 10). W is a small cylinder into 
which the water is admitted through 
the small pipe at the top, and M is a 
large cylinder containing mercury to 
which a glass tube is attached. Be¬ 
fore the water is admitted into W the 
mercury stands at the level of B in 
both the glass tube and large cylinder, 
if the piston does not rest on the 
mercury. When the water is admitted, 
its pressure on the upper end of the piston is pa, if p is the unit-pres¬ 
sure and a the area of the upper end. If A is the area of the lower 
end of the piston, the total pressure upon it is also pa, and hence 
the unit-pressure on the mercury surface is p • a/ A, and this is 
balanced by the column of height z in the glass tube. For example, 
suppose that A = 200a, then the unit-pressure on the mercury sur¬ 
face is 0.005 Pi further, if z be 60 inches, the unit-pressure at B is 
about 2X 14-7 = 2 9-4 pounds per square inch (Art. 4), and accord¬ 
ingly the pressure in W is p = 200 X 29.4 = 5880 pounds per square 
inch, which corresponds to a head of water of about 13 550 feet. 

Prob. 36 . The diameter of the large end of the piston in the last figure 
is 15 inches, and the diameter of the mercury column is I inch. Find the 
distance the piston is depressed when the mercury rises 60 inches. 
































Differential Pressure Gages. Art. 37 


85 


Art. 37 . Differential Pressure Gages 

A differential gage is an instrument for measuring differences 
of heads or pressures, and this must be frequently done in hy¬ 
draulic work. One of the simplest forms is that seen in Fig. 37 a, 
where two water columns from A and D are brought to the sides 
of a common scale upon which the difference of height BC is 
directly read. A better form is one having two glass tubes 




fastened to a scale, these tubes being provided with attachments 
upon which can be screwed the hose leading from the pipe. Where 
it is desired to measure the difference between two large heads, 
provided that this difference is not greater than can be read on 
the scale board, this can be done by connecting the tubes across 
their tops, as in Fig. 376 , and by means of an air pump imposing a 
pressure sufficient to bring the water columns within visible range. 
After this pressure has been imposed the valve at D is closed and 
the difference in the heads read on the scale. 

Fig. 37 c shows the principle of the mercury differential gage* 
Two parallel tubes are open at the top, and here the mercury is 
poured in, the cocks E and F being open and A and C closed; 
the mercury then stands at the same height in each tube. I he 
cocks E and F being now closed and A and C opened, the water 

* For the details of construction see paper by Kuichling in Transac¬ 
tions American Society of Civ il Engineers, 1892, vol. 26, p. 439. 






















































86 


Chap. 4 . Instruments and Observations 


E 


B 


F 
= # 


C 


= # 


Fig. 37c. 


enters at A and C, and the mercury is depressed in one tube and 
elevated in the other. Let the pressure at B be that due to the 
head hi, and the pressure at C be that due to the 
head hi, and let hi be greater than hi ; also let 
the distance read on the scale between the two 
tubes be 2. Then hi= hi + 13-62, or the differ¬ 
ence of the heads of water on B and C is 
h\ — hi = 13.62. Thus if 2 be 1.405 feet, the 
difference of the heads is 19.1 feet. Here, as for 
the mercury gage of Art. 36 , the specific gravity 
of the mercury and water must be known for dif¬ 
ferent temperatures, or comparisons of the instrument with a 
standard gage must be made. 

When the difference of the heads is small, the water gage, 
explained in the first paragraph, cannot measure it with precision, 
especially when the columns are subject to oscillations. To in¬ 
crease the distance between B and C and at the same time decrease 
the amount of oscillation, the oil differential gage, invented by 
Flad in 1885, may be used. Fig. 37 d shows the principle of 
construction.* The cocks A and D being closed and F open, 
sufficient oil is poured in at F to partially fill the 
two tubes. Then F is closed and the water ad¬ 
mitted at A and D , when it rises to B in one 
tube and to C in the other, the oil filling the 
tubes above the water. Let 5 be the specific 
gravity of the water and s' that of the oil, let hi 
be the head of water on B and h 2 that on C, then 
sh 2 = shi + s'z, whence h 2 — h x = (s'/s)z. Kero¬ 
sene oil having a specific gravity of about 0.79 
is generally used, and if the specific gravity of 
the water be unity, the difference of the heads is 0.792. Thus 
2 is greater than h 2 — h x , and hence an error in reading 2 pro¬ 
duces a smaller error in h 2 — h x . The specific gravities of the 
oil and water must be determined, however, so that s'/s can be 



Fig. 37 d. 


*For the details see paper by Williams, Hubbell, and Fenkell in Trans¬ 
actions of American Society of Civil Engineers, 1902, vol. 47, pp. 72-83. 

























Differential Pressure Gages. Art. 37 


87 


expressed to four significant figures when precise work on low 
heads is to be done. 


The difference of head /q — h 2 , determined by these differ¬ 
ential gages, is the difference of the heads due to the pressure at 
the water levels B and C. The difference of the actual heads at 
the points of connection with the pipe under test is next to be 
determined. Fig. 37 e shows a mercury gage set over a water 
pipe for the purpose of determining the loss of head due to a 



valve, the velocity of the water being high, so that the difference 
of pressure at A and D is large. Fig. 37 / shows an oil gage 
set over a similar pipe, the velocity being low, so that the differ¬ 
ence of pressure is small. Let a horizontal plane, represented by 
the broken line, be drawn through the zero of the scale of the gage, 
and let d be the distance of this plane above the horizontal pipe. 
Let b and c be the readings of this scale at the water levels B and 
C in the gage tubes, the difference of these readings being 2. Let 
h x and h 2 be the pressure-heads on B and C, and H 1 and H 2 those 
on A and D. Then H l = h\ -\- b d and H 2 = h 2 + c + d, 
and the difference of these heads is 

Hi — H 2 = h\ — JhArb — c 

which is applicable to both kinds of differential gages. For the 
mercury gage the head hi — h* equals 13*62, while the value of 
b — c is — 2; hence 

Hi — H 2 = 13.62 — 2 = 12.62 

For the oil gage hi - h 2 is - 0.792, while b - c is 2, hence 

Hi — H 2 = — 0.792 + 2 = 0.212 










































88 


Chap. 4 . Instruments and Observations 


In general, if 5' is the ratio of the specific gravity of the mercury 
or oil to that of the water, the difference of the pressure-heads 
at A and D , which is the loss of head due to the valve, is (V — i )z 
for the mercury gage and (i — s')z for the oil gage. 

The principle of the mercury gage can also be applied to the meas¬ 
urement of small differences of head by using a liquid having a specific 
gravity but little heavier than water. Thus Cole, in 1897,* employed 
a mixture of carbon tetrachloride and gasoline which had a specific 
gravity of 1.25; for this mixture Hi — Ho equals 0.25s, or s is four 
times the head Hi — H 2 , and accordingly when II 1 — II 2 is small, the 
error in determining it by the reading s is greatly diminished. 
It may be also noted that when the tube or pipe is not horizontal, the 
expressions (s' — 1 )z and (1 — s)z give the loss of head between the 
two points A and D, although the difference of the actual pressure- 
heads may be greater or less according as A is lower or higher than 
D (Art. 85 ). 

Prob. 37 . In the case of Fig. 37 d let the point D be lower than .4 by 
0.45 foot, and let the reading z be 0.127 foot. How much greater is the 
pressure-head at A than that at D ? 

Art. 38 . Water Meters 

Meters used for measuring the quantity of water supplied 
to a house or factory are of the displacement type; that is, as the 
water passes through the meter it displaces or moves a piston, 
a wheel, or a valve, the motion of which is communicated through 
a train of clock wheels to dials where the quantity that has passed 
since a certain time is registered. There is no theoretical way 
of determining whether or not the readings of the dial hands are 
correct, but each meter must be rated by measuring the discharge, 
in a tank. Several meters may be placed on the same pipe line 
in this operation, the same discharge then passing through each 
of them. When impure water passes through a meter for any 
length of time, deposits are liable to impair the accuracy of its 
readings, and hence it should be rerated at intervals. 

The piston meter is one in which the motion of the water 
causes two pistons to move in opposite directions, the water 

* Transactions American Society of Civil Engineers, 1902, vol. 47, p. 276. 


Water Meters. Art. 38 


89- 


leaving and entering the cylinder by ports which are opened and 
closed by slide valves somewhat similar to those used in the steam- 
engine. The rotary meter has a wheel enclosed in a case so that 
it is caused to revolve as the water passes through. The screw 
meter has an encased helical surface that revolves on its axis 
as the water enters at one end and passes out at the other. The 
disk meter has a wabbling disk so arranged that its motion is 
communicated to a pin which moves in a circle. In all these, and 
in many other forms, it is intended that the motion given to the 
pointers on the dials shall be proportional to the volume of water 
passing through the meter. The dials may be arranged to read 
either cubic feet or gallons, as may be required by the con¬ 
sumers. These meters are of different sizes according to the 
quantity of water to be registered. They all occasion considerable 
loss of head in the pipe on which they are installed and are of 
varying degrees of sensitiveness for small flows. The quantity 
of water registered by a meter of these types varies on account of 
wear both with its age and with the quality of the water it meas¬ 
ures. For these reasons frequent ratings are desirable.* 

The Venturi meter, named after the distinguished hydrauli- 
cian who first experimented on the principle by which it operates, 
was invented by Her- 
schel in 1887.t Fig. 38 a 
shows a horizontal pipe 
having an area tq at 
each end, and the cen¬ 
tral part contracted to 
the area a 2 , with two 
small piezometer tubes into which the water rises. When there 
is no flow, the water stands at the same level in these two 
columns, but when it is in motion, the heights of these columns 
above the axis of the pipe are /q and /q. Let v l and v 2 be the 
mean velocities in the two cross-sections. Then by Art. 24 the 
effective head in the upper section is /q + Vy ; 2g, and that in 

* Transactions American Society of Civil engineers, 1899, vol. 41, and 

Proceedings American Water Works Association, 1910. 

f Transactions American Society of Civil Engineers, 1887, vol. 17, p. 228. 


r vj^- 



Fig. 38a. 











90 


Chap. 4 . Instruments and Observations 


the small section is h 2 + v 2 /2 g ; if there be no losses caused by 
friction, these two expressions must be equal, and hence by the 
theorem of ( 31 ) 2 , 

V 2 2 ~ z>i 2 = 2g(hi — fc) 

Now let Q be the discharge through the pipe, or Q = a 1 v l and also 
Q = a 2 v 2 . Taking the values of and v 2 from these expressions, 
inserting them in the above equation, and solving for Q, gives 

Q = / ■ V 2 g(hi - h) ( 38 ) 

v d\“ CL 2 “ 

which may be called the theoretic discharge. Owing to fric¬ 
tional losses which occur between the two cross-sections, the 
actual discharge q is always less than Q, or q = cQ, in which c 
is a coefficient whose value generally lies between 0.95 and 0.99. 
To determine q, when the coefficient is known, it is hence only 
necessary to measure the difference hi — h 2 , and then compute 
Q by formula ( 38 ). 

The Venturi meter is used for measuring the discharge through 
pipes two inches or more in diameter, the largest meters of 
this type yet undertaken being those for the new Catskill Water 
System of the city of New York. Each of these meters will 
have a capacity of 650 000 000 U. S. gallons per day. They will 
be constructed of reinforced concrete with bronze throat pieces. 
The diameter of each end of the meter tube will be 210 inches, 
while that at the contracted section will be 93 inches. 

The contracted section or throat of the meter is usually made 
from one-quarter to one-ninth of the area of the pipe, and hence 
the velocity through it is from four to nine times that in the pipe. 
The throat area used in any particular case is determined from 
considerations of the various rates of flow to be measured and the 
resulting throat velocities which should not, in order that the 
quantity may be well recorded on the automatic recording ap¬ 
paratus, fall much below 3 feet or far exceed 40 feet per second. 

In practice the two water columns shown on Fig. 38 a may be 
led to a mercury gage, Art. 37 , where the difference between the 
pressure heads hi and h 2 is shown by the difference in level of the 





Water Meters. Art. 38 


91 


two mercury columns. A scale 
graduated so that /q —// 2 varies 
very nearly as q 2 will then 
enable the rate of flow in the 
pipe to be directly read ( 38 ). 
This meter is extensively used 
for the measurement of water 
and other liquids, and its 
capacity and accuracy are 
greater than that of any other 
form yet devised. 

In Fig. 386 is shown a type 
of continuous recording ap¬ 
paratus as constructed by the 
Builders Iron Foundry of 
Providence, R. I., for use with 
the Venturi meter. On the 
upper dial, which is driven by 
a clock, a pen makes on a 
chart a continuous autographic 
record of the rate of flow 
through the meter. By means 
of this chart and a special 
planimeter the quantity of 
water which has passed the 
meter may be determined for 
any desired period. Depend¬ 
ing on the gear of the clock, 
these charts are changed every 
24 hours, every week, or at 
any other desired interval. 
On the central dial the mech¬ 
anism automatically records 
the total quantity of water 
which has passed through the 
meter from the time it was 
set to the time any reading of 



Fig. 38 b. 



































































































92 


Chap. 4 . Instruments and Observations 


the face is taken. On the lower dial the pointer continuously 
indicates the rate of flow, and, depending on the graduations 
of the scale, may indicate in millions of gallons per day, in 
cubic, meters per second, or in any other desired unit. 

A brief description of the operation of this apparatus is as follows. 
The two pressure pipes from the meter tube, Fig. 38 a, are led to two 
mercury chambers connected near their bottoms and so forming a dif¬ 
ferential gage. In each of these chambers is a cast-iron float, and each 
float carries a toothed rack. Each rack meshes with a spur gear, both 
gears being attached to a single shaft which carries the pointer on the 
lower dial. The angular movement of this pointer is therefore exactly 
proportional to any change in the difference of the two mercury 
levels. Attached to this shaft is a cam, the curve of whose face is 
proportional to V hi — h 2 . As the shaft rotates the cam presses 
against and moves a long vertical lever which carries at its top the pen 
which makes the record on the chart on the upper dial. It is evident 
therefore ( 38 ) that the movement of the pen is proportional to q. The 
lever which carries the pen is also connected to a clock-driven in¬ 
tegrating mechanism in a manner such that the speed of the counter 
increases directly as the angular movement of the vertical lever in¬ 
creases from its starting position. The speed of the -counter is at all 
times therefore proportional to the rate of flow through the meter, and 
thus the quantity passing is continuously integrated. The accuracy 
of this recording mechanism can be tested at any time by comparing 
the rate of flow indicated by it with the difference between h x and h 2 
as shown by a differential gage connected to the two pressure tubes 
leading from the meter. A known difference in pressure may also be 
imposed upon the pipes leading to the recording mechanism by means 
of two water columns and the registration of the apparatus observed 
and compared with this known difference. In this way the ap¬ 
paratus can be tested through greater ranges than those usually to 
be obtained under service conditions. 

Another form of recording apparatus for use with the Venturi 
meter is made by the Simplex Valve and Meter Company of 
Philadelphia, Pa.* This apparatus performs all of the functions 
of that above described. Its operation is also based on a cam 
but details of its mechanism are materially different. 


* Proceedings American Water Works Association, 1906. 



Water Meters. Art. 38 


93 


The Premier meter * manufactured by The National Meter 
Company makes use of the Venturi principle though .in a manner 
entirely different from the others above described. It consists 
essentially of a Venturi tube with a by-pass leading from its up¬ 
stream end to its throat. On this by-pass, which is materially 
smaller than the main tube, there is put a displacement meter 
of the piston type which records that proportion of the entire flow 
which passes through it. The ratio between the total flow and 
that indicated by the small meter being determined by experi¬ 
ment, the entire arrangement becomes an instrument for the meas¬ 
urement of water or other liquids. This type of meter is strictly 
of the proportional type, and as such, is open to all of the objec¬ 
tions which hold against the class. It gives best results for throat 
velocities in excess of io feet per second at which the friction 
in the small recording meter becomes relatively small and con¬ 
sequently has less effect on the strict proportionality of flow 
through the two branches. This type of meter is adapted to 
locations close to the hydraulic gradient, where the styles of re¬ 
cording apparatus hereinbefore described could not be used in 
connection with a simple Venturi tube on account of insufficient 
submergence of the throat. For the proper operation of these re¬ 
cording mechanisms it is always necessary that the pressure-head 
at the throat be a positive quantity. 

Still another instrument adapted for making a continuous 
record of the flow of water in a pipe is the Pitotmeter as perfected 
by Cole.f This apparatus consists essentially of a pair of Pitot 
tubes, Art. 41 , which can be inserted through a corporation cock to 
any position within the pipe. One of these tubes looks upstream 
and the other downstream. From them connection is made to 
the branches of a differential gage in which is placed a mixture 
of carbon tetrachloride and gasoline (Art. 37 ). The difference in 
level between the columns is photographically recorded on a strip 
of sensitized paper by means of suitable apparatus, and from this 

* Proceedings American Water Works Association, 1908; Engineer¬ 
ing News, June 16, 1904. 

f Journal New England Water Works Association, 1906; Proceed¬ 
ings American Water Works Association, 1907. 


94 Chap. 4. Instruments and Observations 

recorded difference the quantity of water which has passed through 
the pipe can be computed. With this apparatus the usual 
procedure is to first rate the Pitot tubes (Art. 41 ), and then after 
inserting them into the pipe, making a traverse in order to de¬ 
termine the ratio between the average and maximum velocities. 
This ratio usually varies from o.8o to o.86 (Art. 83 ). Thereafter 
the tubes are set so as to record the maximum velocity, and by 
means of the ratio the average velocity is computed. In order to 
insure correct results the tubes must be carefully rated and care 
be taken to see that they are kept clean of materials deposited 
from the water about their mouths. The Pitotmeter has the 
advantage of causing little or no loss of head. It is a very portable 
instrument, and is particularly adapted for application to water 
waste investigations, pump slippage, and other allied subjects. 

All meters cause a loss in pressure, so that the pressure-head 
in the pipe beyond the meter is less than in the pipe where it 
enters the meter. This is due to the energy lost in overcoming 
friction. For a Venturi tube having a throat area of one-ninth 
that of the pipe the loss of head in feet is about 0.002 iF 2 , where 
V is the velocity in the contracted section in feet per second. 
Thus, when the velocity in a water main is 3 feet per second, the 
velocity in the contracted section will be 27 feet per second, and 
the loss of pressure-head due to the meter tube about 1.53 feet. 

Prob. 38 . A 12-inch pipe delivers 810 gallons per minute through a 
Venturi meter, a 2 being one-ninth of a v Compute the mean velocities 
in the sections a x and a 2 . If the pressure-head in a l is 21.4 feet, compute the 
pressure-head in a 2 . 

Art. 39 . Mean Velocity and Discharge 

In Chap. 3 the velocity of water flowing from an orifice, 
or through a tube or pipe, was regarded as uniform over the 
cross-section. If a is that area, and v the uniform velocity, the 
discharge is q = av ; hence, if a and v can be found by measure¬ 
ment, q is known. In fact, however, the velocity varies in differ¬ 
ent parts of a cross-section, so that the determination of v can¬ 
not be directly made. Yet there always is a certain value for 


Mean Velocity and Discharge. Art. 39 


95 


v, which multiplied into a will give the actual discharge q, and 
this value is called the mean velocity. 

In the case of a stream or open channel the velocity is much 
less along the sides and bottom than near the middle. A rough 
determination of the mean velocity may be made, however, by 
observing the greatest surface velocity by a float, and taking 
eight-tenths of this for the approximate mean velocity. Thus, 
if the float requires 50 seconds to run 120 feet, the mean velocity 
is about 1.9 feet per second ; then if the cross-section be 820 square 
feet, the discharge is 1560 cubic feet per second. 

The practical object of determining the mean velocity is, 
in nearly all cases, to determine the discharge, but as a rule the 
mean velocity cannot be directly observed. A knowledge of 
its value, however, is necessary in all branches of hydraulics, 
since hydraulic coefficients and formulas are based upon it. Ac¬ 
cordingly, many experiments have been made upon small orifices 
and pipes by catching the flow in tanks and thus determining q, 
then the mean velocity has been computed from v = q/a. This 
process has been extended, by indirect methods, to large orifices 
and pipes, and finally to canals and rivers. 

A common method of finding the discharge of a stream is 
to subdivide the cross-section into parts and determine their 
areas a u a 2 , etc., the sum of which is the total area a. Then, 
if Vi, v 2 , etc., are the mean velocities in these areas, and if these 
are determined by observations, the discharge is 

q = aiVi -f a 2 v 2 + (13V 3 + etc. ( 39 ) 

Here the mean velocities may be roughly found by observing 
the passage of a surface float at the middle of each subdivision 
and multiplying this surface 
velocity by 0.9. There 
are, however, more precise 
methods, one of which will 
be explained in Art. 40 , 
while others will be described in Chap. 10 . When q has been 
found in this manner, the mean velocity of the stream may be 
computed, if desired, by v = q/a. 
























96 


Chap. 4. Instruments and Observations 


Formula ( 39 ) applies also to a cross-section of any kind. 
Thus, let the pipe of Fig. 396 be divided by concentric circles 

into the areas, a l5 a 2 , a 3 , a 4 , and let the mean 
velocities v 2 , v 3 , i> 4 , be determined by obser¬ 
vation for each of these areas; the discharge 
q is then given by ( 39 ). Again, in the con¬ 
duit of Fig. 126 a, let a velocity observation be 
taken at each of the 97 points marked by a 
dot, these points being uniformly spaced over 
the cross-section, so that each of the areas a x , a 2 , etc., may be 
regarded as 9V a. Then from ( 39 ) the discharge is 

q = 9V g(®i + ^2 + v 3 + • •• + ^ 97 ) = av 



or v is the sum of the individual velocities divided by 97. In 
general, if a cross-section be divided into n equal parts, the mean 
velocity is the average of the n observed velocities. This result 
is the more accurate the greater the number of parts into which 
the cross-section is divided. If the number of parts be infinite 
and the water passing through each be called a filament, the mean 
velocity in the cross-section may be defined as the average of the 
velocities of all the filaments. 

Prob. 39. A water pipe, 3 inches in diameter, is divided into three parts 
by concentric circles whose diameters are 1,2, and 3 inches. The mean veloc¬ 
ities in these parts are found to be 6.6, 4.8, and 3.0 feet per second. Com¬ 
pute the discharge and mean velocity for the pipe. 


Art. 40 . The Current Meter 

In 1790 the German hydraulic engineer Woltmann invented 
an apparatus for measuring the velocity of flowing water which 
was later improved by Darcy and others, and is now extensively 
used for gaging streams and other open channels. This meter 
is like a windmill, having three or more vanes mounted on a 
spindle and so arranged that the face of the wheel always stands 
normal to the direction of the current, the pressure of which causes 
it to revolve. The number of revolutions of the wheel is approxi¬ 
mately proportional to the velocity of the current. In the best 
forms of this instrument the number of revolutions made 


The Current Meter. Art. 40 


97 


in a given time is determined and recorded by an apparatus 
placed near the observer on a bridge, in a boat, or elsewhere. 
In these forms an electric connection is made and broken at every 
fifth revolution and a dial on the recording apparatus affected. 
By means of a telephone receiver the making and breaking of 
the circuit can be made audible to the observer, who in such 
case simply keeps count of the number of clicks and observes on 
a stop-watch the time elapsed for a given number of revolutions. 



The meter may be operated by placing it on a rod on which 
its position may be changed at will or by suspending it from a 
chain or rope. The former of these methods is applicable only 
to small streams and to cases where the velocity is low. Under 
the second method the meter can best be operated from a bridge, 
and in some cases at permanent gaging stations in lieu of a bridge 
a wire cable may be stretched across the stream and at a sufficient 
height above it, so that 
the operator, when 
seated in a cage which 
travels on the cable, 
will have room for 
operation. On very 
large streams or where 
the expense of a cable 
is not warranted the 
gagings may be made 
from a boat. At times 
of low water, in shal¬ 
low streams the meter 
is carried and held di- 


Fig. 406 . 


Fig. 40 a. 

































































98 


Chap. 4. Instruments and Observations 


rectly in position by the observer who wades out into the 
stream. In such cases care must be taken to hold the meter 
clear of the disturbing influence of the observer’s presence. 

Figure 40 a shows the recording dial of an electrically operated de¬ 
vice for counting the revplutions of a meter, and in Fig. 40 & is shown 
the Price current meter, a form extensively used in the United States. 
The cups or vanes are kept facing the current by means of the cross¬ 
shaped rudder immediately behind them. At the lower end of the 
standard is a heavy torpedo-shaped lead weight also equipped with 
rudder vanes. The supporting cable is shown connected to the upper 
end of the standard by a snap, and the electric connection wires are 
shown extending from the battery in the leather case through the meter 
and thence to the telephone receiver. Both the battery and the re¬ 
ceiver are carried by the observer. In order to assist in keeping the 
meter more nearly vertical in swiftly flowing streams a line may be 
attached to the supporting cable a short distance above the meter and 
carried to some point upstream, so that a pull on it will help to make the 
meter better maintain its position. 

A current meter cannot be used for determining the velocity 
in a small trough or channel, since the introduction of it into the 
cross-section would contract the area and cause a change in the 
velocity of the flowing water. In large conduits, canals, and rivers 
it is, however, a convenient and accurate instrument. By simply 
holding it at a fixed position below the surface the velocity at 
that point is found; by causing it to descend at a uniform rate 
from surface to bottom the mean velocity in that vertical is 
obtained; and by passing it at a uniform rate over all parts of 
the cross-section of a channel the mean velocity v can be directly 
determined. This latter procedure is one which can be put into 
practice only in small channels and under unusual conditions. 
It is mentioned here simply to illustrate the various uses to 
which the current meter may be put. 

In operation the current meter is generally suspended from 
a cable which is graduated so that the distance of the cen¬ 
ter of the meter below the surface of the water can be directly 
read by the observer. The current meter, like every other 
instrument, must be used and handled with care to produce 


The Current Meter. Art. 40 


99 


the best results. Hoyt * has well summarized recent current 
meter practice and the results which have been obtained. 

To derive the velocity of the water from the number of 
recorded revolutions per second the meter most first be rated 
by pushing it at a known velocity through still water. The 
best place for doing this is in a pond or navigation canal, where 
the water has no sensible velocity. A track is built along the 
bank on which a small car can be moved at a known velocity. 
From this car the meter is suspended into the water either 
from a rod or a cable, and the method of suspension used should 
be the same as that to be employed in actual service. The lowest 
velocity of the car should be that at which the meter will just start 
and continue revolving; this velocity is from o.i to 0.2 feet per 
second. The highest velocity should be somewhat in excess 
of the actual velocities to be observed, and ratings are usually 
carried up to velocities of from 10 to 15 feet per second. It 
is always found that the number of revolutions per minute 
is not exactly proportional to the velocity of the car, and hence 
when the meter is held stationary in running water, the velocity 
of the water is not proportional to the number of revolutions. 

From the observations made at the different known velocities 
there is prepared a rating table showing the velocity of the water 
in feet per second corresponding to the number of meter revo¬ 
lutions. This form of table is best, since in making observations 
best results are obtained by noting the number of seconds required 
to complete a certain number of revolutions. To make such a 
table the known velocities of the car are taken as abscissas 
on cross-section paper and the number of revolutions as ordinates, 
and a point corresponding to each observation is plotted. A mean 
curve may then be drawn to agree as closely as possible with the 
plotted points, and from this curve the velocity corresponding to 
any number of revolutions can be taken off. This curve may be 
expressed by an equation of the form V — a + bn or V — a -{-bn 
+ cn 2 , in which V is the velocity of the car in feet per second and 
n in the number of revolutions of the meter per second. By the 

* Transactions American Society of Civil Engineers, 1910, vol. 66, p. 70. 


100 


Chap. 4. Instruments and Observations 


aid of the Method of Least Squares the constants of the equation 
may then be computed and the curve determined (Art. 42 ). 
In the case of the small Price meter it has been found that the 
curve is very closely approximated by two straight lines AB and 
BC, as shown in Fig. 40 c, which is a typical rating curve for this 



Fig. 40r. 


type of meter.* This curve was based on thirty-five observa¬ 
tions at different velocities, and practically all of them fell on 
the line ABC which is also very nearly a straight line. 

An examination | of the rating tables of a number of meters 
has shown that possible errors due to differences in rating are 
quite small, and that a Price meter in good condition can be used 
with a standard rating table without serious error for all veloc¬ 
ities greater than 0.5 foot per second and then generally within 
about 2 percent. 

While the current meter is an extensively used instrument, there 
are, as in most other hydraulic work, certain features which are not 
yet fully understood. These are the differences shown in the results of 
the ratings of the same meter when held on a rod and when suspended 
by a cable. J It has also been found that the rating of a meter made in 
still water differs somewhat from that made in running water ,t but 
no successful means for making direct running water ratings have 
as yet been devised. Many good comparisons between current meter 
gagings and weir measurements have been made, but the current meter 

* Transactions American Society of Civil Engineers, 1910, vol. 66, p. 83. 

t Transactions American Society of Civil Engineers, 1910, vol. 66, p. 83. 

t Water Supply and Irrigation Paper, No. 95, U. S. Geological Survey. 

c < 

1 V ( 



























The Pitot Tube. Art. 41 


101 


velocities in all of them have been relatively low, so that no complete 
comparison has up- to the present been possible. 

Prob. 40. In order to rate a certain current meter, three observations 
were taken in still water, as follows: 

Velocity of the car = 2.0 3.8 7.4 feet per second 

Revolutions per minute = 30 60 120 

Plot these observations on cross-section paper and deduce, without using 
the Method of Least Squares, the relation between V and n in the equation 
V = a + bn. 


Art. 41 . The Pitot Tube 

About 1750 the French hydraulic engineer Pitot invented a 
device for measuring the velocity in a stream by means of the 
velocity-head which it will produce. In its simplest form it 
consists of a bent tube, the mouth of which is placed so as to 
directly face the current. The water then rises in the vertical 
part of the tube to a height h above the surface of the flowing 
stream, and this height is equal to the velocity-head v 2 / 2 g, so that 
the actual velocity v is in practice approximately equal to V2 gh. 
As constructed for use in 
streams, Pitot’s appa¬ 
ratus consists of two 
tubes placed side by side 
with their submerged 
mouths at right angles, 
so that when one is op¬ 
posed to the current, as 
seen in Fig. 416 , the other stands normal to it, and the water 
surface in the latter tube hence is at the same level as that of 
the stream. Both tubes are provided with cocks which may 
be closed while the instrument is immersed, and it can be then 
lifted from the water and the head h be read at leisure. It is 
found that the actual velocity is always less than Vzg/q and 
that a coefficient must be deduced for each instrument by mov¬ 
ing it in still water at known velocities. Pitot’s tube has the 
advantage that no time observation is needed to determine the 
velocity, but it has the disadvantage that the distance h is 























102 Chap. 4. Instruments and Observations 

usually very small, so that an error in reading it has a large 
influence. Although the instrument was improved by Darcy in 
1856 and used by him for some stream measurements, it was for 
a long time regarded as having a low degree of precision. 

When using a Pitot tube for measuring the velocity in a 
stream, the two columns maybe raised above the level of the water 
in the stream and brought to a height convenient for observation 
by partly exhausting the air from the tubes above the columns. 
This procedure is analogous to the imposing of an air pressure 
above the water columns in the case of high heads, as was de¬ 
scribed in Art. 37 . 

In 1888 Freeman made experiments on the distribution of 
velocities in jets from nozzles, in which an improved form of 
Pitot tube was used.* The point of the tube facing the current 
was the tip of a stylographic pen, the diameter of the opening 
being about 0.006 inch. This point was introduced into differ¬ 
ent parts of the jet and the pressure caused in the tube was meas¬ 
ured by a Bourdon pressure gage reading to single pounds. 
The velocities of the jets were high; for example, in one series 
of observations on a jet from a if-inch nozzle, the gage pressures 
at the center and near the edge were 51.2 and 18.2 pounds per 
square inch, which correspond to velocity-heads of 118.2 and 42.0 
feet, or to velocities of 87.2 and 52.0 feet per second. By com¬ 
puting the mean velocity of the jet from measurements in con¬ 
centric rings (Art. 39 ) and also from the measured discharge, 
Freeman concluded that any velocity as determined by the tube 
was smaller than that computed from v = V2 gh by less than one 
percent. This investigation established the fact that the Pitot 
tube is an instrument of great precision for the measurement 
of high velocities. 

Experiments on the flow of water in pipes, in which Pitot 
tubes were successfully used, were made in 1897 by Cole at Terre 
Haute, and in 1898 by Williams, Hubbell, and Fenkell at Detroit.t 
In the Detroit experiments the tube was introduced into the pipe 

* Transactions American Society of Civil Engineers, 1889, vol. 21, P- 4 X 3- 

f Transactions American Society of Civil Engineers, 1902, vol. 47, pp. 12, 275. 



The Pitot Tube. Art. 41 


103 


through an opening provided with a stuffing-box, so that the 
point of the tube might be placed at any desired position. The 
tubes had openings at their points A inch in diameter and other 
openings of the same size on their sides to admit the static pres¬ 
sure of the water. These latter openings led to a common chan¬ 
nel parallel to that leading from the point, and each of these was 
connected to a rubber hose running to a differential gage, con¬ 
sisting of two parallel glass tubes open at the top, where the dif¬ 
ference of head was read on a scale. In order to be able to deduce 
the velocities in the pipe from the readings of the gage, the Pitot 
tubes were rated by moving them in still water at known veloc¬ 
ities as for the current meter (Art. 40 ). Thus a coefficient c was 
derived for each tube for use in the formula v = c^/2gh. This 
coefficient was found to range from o.86 to 0.95 for different tubes, 
and it varied but little with v. 

Many different forms of Pitot tubes have been made and experi¬ 
mented upon. Each of these forms has, in common with the others, 
the pressure opening which faces the current, though the shape and 
dimensions of this opening differ materially in the various types. In 
some of them the static pressure is admitted through a hole in the side 
of the apparatus, while in others it is admitted through a number of 
such holes. In another type the tube is made symmetrical with an 
opening looking downstream. In this case the water column connected 
with the upstream opening will indicate the velocity head, while that 
connected with the opening which faces downstream will indicate 
a pressure less than the static head on account of the negative head 
induced by the arrangement. The difference between the two columns 
is thus increased and its reading on the scale rendered more easy, while 
the proportional error of any reading is also reduced. In Fig. 41 c is 
shown a form of tube used by the U. S. Geological Survey* for the meas¬ 
urement of velocity in small and shallow streams in connection with 
experiments on the transporting capacity of currents, while in Fig. 4 Id 
is shown the type used in connection with the Pitotmeter (Art. 38 ). 
In this figure is shown also the method of introducing the tubes into 
a pipe where the velocity is to be measured. 

Some recent comparisons * between the still and moving water 
ratings of Pitot tubes indicate that there may be a difference between 


* Engineering News, Aug. 12, 1909. 



104 


Chap. 4 . Instruments and Observations 


the results obtained by these two methods. It is desirable, of course, 
that every instrument should be rated under conditions similar to those 
in which it is to be used. One of the ways of rating a Pitot tube 



in running water is that suggested and used by Judd and King* 
who placed the tube used by them at the contracted section of a jet 
and concluded that its coefficient was i.oo. 

Prob. 41 . Explain how a well-rated Pitot tube may be used to measure 
the speed of a boat or ship. 

Art. 42 . Discussion of Observations 

An observation is the recorded result of a measurement. All 
measurements are affected with errors due to imperfections of 
the instrument and lack of skill of the observers, and the recorded 
results contain these errors. Thus, if 6.05, 6.02, 6.01, and 6.04 
inches be four observations on the diameter of an orifice, all of 

* Engineering News, Sept. 27, igo6. 





















































Discussion of Observations. Art. 42 105 

these cannot be correct, and probably each is in error. The best 
that can be done is to take the average of these observations, or 
6.03 inches, as the most probable result, and to use this in the 
computations. 

An observer is often tempted to reject a measurement when 
it differs from others, but this can only be allowed when he is 
convinced that a mistake has been made. A mistake is a large 
error, due generally to carelessness, and must not be confounded 
with the small accidental errors of measurement. When a series 
of observations is placed before a computer, he should never be 
permitted to reject one of them, unless there is some remark in 
the note-book which casts doubt upon it. 

Graphical methods of discussing and adjusting observations, 
like that mentioned in Art. 40 , are of great value in hydraulic 
work. As another example, the following observations made by 
Darcy and Bazin on the flow of water in a rectangular trough, 
1.812 meters wide and having the uniform slope 0.049, ma Y be 
noted. Water was allowed to run through it with varying depths, 
and for each depth the mean velocity (Art. 39 ) and the hydraulic 
mean depth (Art. 112 ) was determined by measurement. Let 
v be the mean velocity and r the hydraulic mean depth; then five 
measurements gave the following observations, v being in meters 
per second and r in centimeters. Let it be assumed that the 

No. = 1 2 3 4 5 

v= 1.73 1.98 2.17 2.33 2.46 

r=n.4 14.4 17.0 19.2 21.2 

relation between v and r is of the form v = mr n , and let it be re¬ 
quired to determine the most probable values of m and n. 

For each of these observations a point may be plotted on cross- 
section paper, taking the values of v as ordinates and those of r 
as abscissas, and a smooth curve may then be drawn so as to agree 
as nearly as possible with the points. Such a curve, however, 
is of little assistance in determining the values of m and n , unless 
the curve should be a straight line drawn through the origin, in 
which case it is plain that n is unity and that m is the tangent ol 


106 Chap. 4 . Instruments and Observations 


the angle that the line makes with axis of abscissas. In this case 
no straight line can be drawn approximating to the points and 
passing through the origin, but the plot gives the curve shown 
in Fig. 42 a. If, however, the logarithm of each side of the as¬ 
sumed formula be taken, it becomes 

log v = n log r + log m 

which represents a straight line if log v be considered as the 
variable ordinate and log r as the variable abscissa, log m being 


ec 

<w 

3 

a 




1 2 r 

0 4 

0 _-©•- 

V- 












0 5 10 15 20 25 

Values of r 

Fig. 42a. 


the intercept on the axis of ordinates and n the tangent of the 
angle which the line makes with the axis of abscissas. On plot¬ 
ting the points corresponding to the values of log v and log r, it is 
seen that a straight line can be drawn closely agreeing with the 


5 



Fig. 426. 


points, that this line cuts the axis of ordinates at a distance of 
about 0.35 below the origin, and that the tangent of the angle 
made by it with the axis of abscissas is about 0.55. Hence (Fig. 
42 b) n = 0.55, log m = — 0.35 = 1.65, or m = 0.446; then 

logy = 0.55 log r — 0.35 or y = o.446r 055 

is an empirical formula for computing the mean velocity in this 
trough. Using the above values of r and computing those of v, 
it is found that the computed and observed results agree fairly, 



































Discussion of Observations. Art. 42 


107 


the former being generally a little smaller, which is due to the 
fact that only two significant figures are found from the plot. 

Whenever a series of plotted points can be closely represented 
by a straight line on logarithmic section paper, the equation be¬ 
tween the variables is an exponential one. Numerous exponential 
formulas for the flow of water in pipes and channels rest upon 
the judgment of the investigator in deciding that the plotted 
points are sufficiently well represented by a straight line. 

There is a process, known as the Method of Least Squares, by 
which the constants of an empirical formula may be obtained from ob¬ 
servations with a higher degree of precision than by any graphic method. 
Its application to the above case will here be given. Let the simul¬ 
taneous values of log v and log r for each experiment be placed in 
the logarithmic formula as follows: 


for No. i, 
for No. 2, 
for No. 3, 
for No. 4, 
for No. 5, 


0.238 = 1.057 n + log m 
0.297 = 1.158ft + log w 
0.336 = 1.230ft + log m 
0.367 = 1.283ft + log m 
0.391 = 1.326ft + log m 


These five equations contain two unknown quantities, n and log m, 
but no values of these can be found that will exactly satisfy all the 
equations. The best that can be done is to find the values that have 
the greatest degree of probability, and these will satisfy the equations 
with the smallest discrepancies. To do this, let each equation be 
multiplied by the coefficient of n in that equation and the results be 
added; also let each equation be multiplied by the coefficient of log m 
in that equation and the results be added. Thus are found the two 
normal equations containing the two unknown quantities: 

1.998 = 7.375ft + 6.054 log m 
1.629 = 6.054ft + 5.000 log m 

and the solution of these gives n = 0.571 and log m — — 0.366. 
Since — 0.366 equals 1.634, the value of m is 0.431, and then 

log v = 0.571 log r — 0.366 or z; = 0.431 r 0bn 
is the empirical formula for this particular case. 

The Method of Least Squares is usually more laborious than the 
graphical method, but it has the great advantage that its results are 


108 


Chap. 4 . Instruments and Observations 


the most probable ones that can be derived from the given data. It 
has the further advantage that all computors will derive the same 
results, whereas in the graphic method the results will usually differ, 
because the position of the line drawn on the plot is affected by the 
different degrees of judgment and experience of the draftsmen. It 
will be seen from Fig. 42 b that it is not very easy to determine close 
values of log m since the plotted points are so far away from the origin. 

Prob. 42a. In order to rate a certain current meter four observations 
were taken in still water as follows: 

Velocity of the car 0.7 2.4 4.7 9.3 feet per second 

Revolutions of meter 18 60 120 240 per minute 

Find the values of a and b in the formula v = a + bn, both by plotting and 
by die method of least squares. 

Prob. 42fr. Three observations of horizontal angles are made at the 
station O, which give .105 = 62°i7', BOC = 20°$$', A 0 C = 82°SS'• Ad¬ 
just these observations by the method of least squares so that the large 
angle may be equal to the sum of its parts. 


Standard Orifices. Art. 43 


109 


CHAPTER 5 


FLOW OF WATER THROUGH ORIFICES 


Art. 43. Standard Orifices 


Orifices for the measurement of water are usually placed in 
the vertical side of a vessel or reservoir, but may also be placed 
in the base. In the former case it is understood that the upper 
edge of the opening is completely covered with water; and gen¬ 
erally the head of water on an orifice is at least three or four times 
its vertical height. The term “standard orifice” is here used to 
signify that the opening is so arranged that the water in flowing 
from it touches only a line, as would be the case in a plate of no 
thickness. To secure this result the inner part of the opening is 
a definite edge, which alone is touched by the water. In pre¬ 
cise experiments the orifice may be in a metallic plate whose 
thickness is really small, as at A in the figure, but more commonly 
it is cut in a board or plank, care being taken that the inner edge 
is sharp and definite. It is usual to bevel the outer part of the 
orifice, as at C, so that the escaping jet 
may by no possibility touch the same 
except at the inner edges. The term 
“orifice in a thin plate” is often used 
to express the condition that the water 
shall only touch the edges of the open¬ 
ing along a line. This arrangement 
may be regarded as a kind of standard 
apparatus for the measurement of 
water; for, as will be seen later, the discharge is modified when 
the inner edges are rounded, and different degrees of rounding 
give different discharges. The standard arrangements shown in 
Fig. 43a are accordingly always used when water is to be meas¬ 
ured by the use of orifices. 



I 
pf 

m 






in 


i 


Unk 


I, 


B 


Fig. 43a. 


W/A 





























110 


Chap. 5. Flow of Water through Orifices 


The contraction of the jet which is always observed when 
water issues from a standard orifice, as described above, is a most 
interesting and important phenomenon. It is due to the circum¬ 
stance that the particles of water as they approach the orifice 
move in converging directions, and that these directions continue 
to converge for a short distance beyond the plane of the orifice. 
It is this contraction of the jet that causes only the inner corner 
of the orifice to be touched by the escaping water. The appear¬ 
ance of such a jet under steady flow, issuing from a circular ori¬ 
fice, is that of a clear crystal bar whose beauty claims the ad¬ 
miration of every observer. The convergence due to this cause 
ceases at a distance from the plane of the orifice of about one-half 
its diameter. Beyond this section the jet enlarges in size if it be 
directed upward, but decreases in size if it be directed downward 
or horizontally. 

The contraction of the jet is also observed in the case of rec¬ 
tangular and triangular orifices, its cross-section being similar 

to that of the orifice until the 
place of greatest contraction is 
passed. Fig. 436 shows in the top 
row cross-sections of a jet from a 
square orifice, in the middle row 
those from a triangular one, and 
in the third row those from an 
elliptical orifice. The left-hand 
diagram in each case is the cross- 
section of the jet near the place 
of greatest contraction, while.the following ones are cross-sec¬ 
tions at greater distances from the orifice, and the jets are sup¬ 
posed to be moving horizontally or nearly so. 


□ o * 

A O V 




0 


O O 

Fig. 43 b. 


Owing to this contraction, the discharge from a standard 
orifice is always less than the theoretic discharge, which, from 
Arts. 22 and 30, would be expressed by 

Q = a V 2gh (43) 

where a is the area of the orifice and h the head above its center. 
It is evident that the quantity of water passing the plane of the 


Coefficient of Contraction. Art. 44 


111 


orifice and that passing the plane of the contracted section in any 
unit of time are the same, and since there probably can be no 
appreciable change in the density of the water, there must there¬ 
fore be an increase in velocity between these two planes. The 
reasons for such an increase are not fully known. It is not prob¬ 
able that the velocity at the center of the jet changes materially, 
but rather that the increase occurs in its outer filaments, so that at 
the contracted section they are all traveling parallel with each 
other and at the same velocity.* 

It is the object of this chapter to determine how the theoretic 
formulas for orifices given in Chap. 3 are to be modified so that 
they may be used for the practical purposes of the measurement 
of water. This is to be done by the discussion of the results of 
experiments. It will be supposed, unless otherwise stated, that 
the size of the orifice is small compared with the cross-section of 
the reservoir, so that the effect of velocity of approach may be 
neglected (Art. 24). 

Prob. 43. At a distance from a circular orifice of one-half its diameter 
a jet has a diameter of i inch and a velocity of 16 feet per second. When it 
is directed vertically downward, what is the diameter of a section 5 feet 
lower? When it is directed vertically upward, what is the diameter of a 
section 5 feet higher? 

Art. 44. Coefficient of Contraction 

The coefficient of contraction is the number by which the area 
of the orifice is to be multiplied in order to give the area of the 
section of the jet at a distance from the plane of the orifice of 
about one-half its diameter. Thus, if c' be the coefficient of con¬ 
traction, a the area of the orifice, and a' the area of the contracted 

section of the jet, then , , (AA \ 

J a =c a (44) 

The coefficient of contraction for a standard orifice is evidently 
always less than unity. 

The only direct method of finding the value of c' is to measure 
by calipers the dimensions of the least cross-section of the jet. 
The size of the orifice can usually be determined with precision, 


* Engineering News, Sept. 27, 1906. 


112 Chap. 5 . Flow of Water through Orifices 

and with care almost an equal precision in measuring the jet. To 
find c' for a circular orifice let d and d ' be the diameters of the 
sections a and a' ; then 

c = a /a = ( d r /d) 2 

Therefore the coefficient of contraction is the square of the ratio 
of the diameter of the jet to that of the orifice. The first meas¬ 
urements were made by Newton * who found the ratio of d' to d 
to be 21/25, which gives for c the value 0.73. The experiments of 
Bossut gave from 0.66 to 0.67 ; and Michelotti found from 0.57 
to 0.624 with a mean of 0.61. Eytelwein gave 0.64 as a mean 
value, and Weisbach mentions 0.63. 

The following mean value will be used in this book, and it 
should be kept in mind by the student: 

Coefficient of contraction c =0.62 

or, in other words, the minimum cross-section of the jet is 62 per¬ 
cent of that of the orifice. This value, however, undoubtedly 
varies for different forms of orifices and for the same orifice under 
different heads, but little is known regarding the extent of these 
variations or the laws that govern them. Probably c' is slightly 
smaller for circles than for squares, and smaller for squares than 
for rectangles, particularly if the height of the rectangle is long 
compared with its width. Probably also c' is larger for low 
heads than for high heads. 

Judd and King in 1906,! using a specially constructed pair of 
calipers, { found the following values for the coefficient of con¬ 
traction for standard orifices : 

Orifice diameter, inches, 0.75 1.00 1.50 2.00 2.50 

Coefficient of contraction, 0.6134 0.6115 0.6051 0.6082 0.5955 

Prob. 44 . The diameter of a circular orifice is 1.905 inches. Three 
measurements of the diameter of the contracted section of the jet gave 1.55, 
1.56, and 1.59 inches. Find the mean coefficient of contraction. 

* Philosophise Naturalis Principia Mathematica, 1687, Book II, prop. 36. 

f Engineering News, Sept. 27, 1906. f Science, March 4, 1904. 


Coefficient of Velocity. Art. 45 


113 


Art. 45 . Coefficient of Velocity 


The coefficient of velocity is the number by which the theoretic 
velocity of flow from the orifice is to be multiplied in order to give 
the actual velocity at the least cross-section of the jet. Thus, if 
Ci be the coefficient of velocity, V the theoretic velocity due to the 
head on the center of the orifice, and v the actual velocity at the 
contracted section, then 



( 45 ) 


The coefficient of velocity must be less than unity, since the 
force of gravity cannot generate a greater velocity than that due 
to the head. 

The velocity of flow at the contracted section of the jet cannot 
be directly measured. To obtain the value of the coefficient of 
velocity, indirect observations have been taken on the path of the 
jet. Referring to Art. 25 , it will be seen that when a jet flows 
from an oriflee in the vertical side of a vessel, it takes a path whose 
equation is y = gx 2 /2V 2 , in which .r and y are the coordinates 
of any point of the path measured from vertical and horizontal 
axes, and v is the velocity at the origin. Now placing for v its 
value c x V 2 gh , and solving for c lf gives 



Therefore c x becomes known by the measurement of the head h 
and the coordinates x and y. In making this experiment it would 
be well to have a ring, a little larger than the jet, supported by a 
stiff frame which can be moved until the jet passes through the 
ring. The flow of water can then be stopped, and the coordinates 
of the center of the ring determined. By placing the ring at 
different points of the path different sets of coordinates can be 
obtained. The value of x should be measured from the contracted 
section rather than from the orifice, since v is the velocity at the 
former point and not at the latter. 

By this method of the jet Bossut in two experiments found for 
the coefficient of velocity the values 0.974 and 0.980, Michelotti 
in three experiments obtained 0.993, 0.998, and 0.983, and Weis- 
bach deduced 0.978. Great precision cannot be obtained in these 



114 


Chap. 5 . Flow of Water through Orifices 


determinations, nor indeed is it necessary for the purposes of 
hydraulic investigation that c Y should be accurately known for 
standard orifices. As a mean value the following may be kept 

in the memory: Coefficient of ve i oci t y Cl = 0.98 

or, the actual velocity of flow at the contracted section is 98 per¬ 
cent of the theoretic velocity. The value of c 1 for the standard 
orifice is greater for high than for low heads, and may probably 
often exceed 0.99. 

Another method of finding the coefficient c Y is to place the 
orifice horizontal so that the jet will be directed vertically up¬ 
ward, as in Fig. 22 . The height to which it rises is the velocity- 
head h Q = v 2 / 2 g, in which v is the actual velocity c Y ’y/2gJi. Accord¬ 
ingly, h 0 = c 2 h , from which c Y may be computed. For example 
if, under a head of 23 feet, a jet rises to a height of 22 feet, the 
coefficient of velocity is 

Ci = V h 0 /h = V22/23 = 0.978 

This method, however, fails to give good results for high veloci¬ 
ties, owing to the resistance of the air, and moreover it is impossi¬ 
ble to measure with precision the height h 0 . 

For a vertical orifice Poncelet and Lesbros found, in 1828, 
that the coefficient c Y was sometimes slightly greater than unity, 
and this was confirmed by Bazin in 1893. This is probably 
due to the fact that the head is greater for the lower part of the 
orifice than for the upper part, and hence V2 gli does not represent 
the true theoretic velocity. The same experimenters found no 
instance of a horizontal orifice where the coefficient exceeded unity. 


Since the coefficient of velocity is the ratio between the coefficient 
of discharge (Art. 46 ) and the coefficient of contraction, it may be 
computed from observations on these quantities. Thus Judd and 
King,* using the average of the coefficients of contraction shown 
in Art. 44 and the average of the coefficients of discharge shown in 
Art. 46 , found the following: 


coefficient of velocity 


coefficient of discharge _ 0.60664 
coefficient of contraction o. 60674 


0.99983 


* Engineering News, Sept. 27, 1906. 








Coefficient of Discharge. Art. 46 


115 


By traversing the jets with a Pitot tube they also determined the co¬ 
efficient of velocity to be 0.99993 and showed that the velocity at 
the contracted area is uniform throughout its cross-section. From 
the results of these experiments they concluded that the coefficient 
of velocity is unity and hence adopted the term “frictionless orifice ” 
as descriptive of the particular standard orifices used by them. 

Prob. 45 . The range of a jet is 13.5 feet on a horizontal plane 2.82 feet 
below the orifice which is under a head of 14.38 feet. Compute the coeffi¬ 
cient of velocity. 

Art. 46 . Coefficient of Discharge 

The coefficient of discharge is the number by which the theo¬ 
retic discharge is to be multiplied in order to obtain the actual 
discharge. Thus, if c is the coefficient of discharge, Q the theo¬ 
retical, and q the actual discharge per second, then 

q = cQ ( 46 ) 1 

Here also the coefficient c is a number less than unity. 

The coefficient of discharge can be accurately found by 
allowing the flow from an orifice to fall into a vessel of constant 
cross-section and measuring the heights of water by the hook gage 
(Art. 35 ). Thus q is known, and Q having been computed, 

c = q/Q ( 46) 2 

For example, a circular orifice of 0.1 foot diameter was kept un- 
deraconstant head of 4.677 feet; during 5 minutes and 32^ seconds 
the jet flowed into a measuring vessel which was found to contain 
27.28 cubic feet. Here the actual discharge was 

q = 27.28/332.2 = 0.08212 cubic feet per second 

The theoretic discharge, from formula ( 30 ), is 

Q = 7T X 0.05 2 X 8.02 V4.677 = 0.1361 cubic feet per second 

Then the coefficient of discharge is found to be 

c = 0.08212/0.1361 = 0.604 

In this manner thousands of experiments have been made upon 
different forms of orifices under different heads, foi accurate 
knowledge regarding this coefficient is of great importance in 
practical hydraulic work. 



116 


Chap. 5 . Flow of Water through Orifices 


The following articles contain values of the coefficient of 
discharge for different kinds of orifices, and it will be seen that 
in general c is greater for low heads than for high heads, greater 
for rectangles than for squares, and greater for squares than for 
circles. Its value ranges from 0.59 to 0.63 or higher, and as a 
mean to be kept in mind the following value may be stated : 

Coefficient of discharge c — 0.61 

or, the actual discharge from a standard orifice is, on the average, 
about 61 percent of the theoretic discharge. 

The coefficient c may be expressed in terms of the coefficients 
c' and Ci. Let a and a' be the areas of the orifice and the cross- 
section of the contracted jet, and Q and q the theoretic and actual 
discharge per second. Then, since a'/a = c 

q_ a C\ v 2 gh _ a r _ r , r 

Q a 

and therefore the coefficient of discharge is the product of the 
coefficients of contraction and velocity. 

The coefficient of discharge is of greater importance than the 
coefficients of contraction and velocity, since it is the quantity 
generally used in making measurements of water. Tabulations 
of its values for all practical cases are given below. 

Prob. 46 . The diameter of a contracted circular jet was found to be 
0.79 inches, the diameter of the orifice being 1 inch. Under a head of 16 
feet the actual discharge per minute was found to be 6.42 cubic feet. Find 
the coefficient of velocity. 

Art. 47 . Circular Vertical Orifices 

Let a circular orifice of diameter d be in the side of a vessel 

and let h be the head of water on its center. Then, from Art. 

22 , the theoretic mean velocity is V2 gh, and from Art. 30 the 

theoretic discharge is _ . ,—7 

Q = i Trd~ V 2 gh 

which applies when h is large compared with d. 

To deduce a more exact formula let the radius of the circle 
be r, and let an elementary strip be drawn at a distance y above 







Circular Vertical Orifices. Art. 47 


117 


the center ; the length of this is 
2 Vr— its area is 2 fry Vr— y 2 , and 
the head upon it is h — y. Then the 
theoretic discharge through this strip 

* S BQ = 2 By V r 2 — y 2 V 2 g{h — y) 

To integrate this (h — y ) 2 is to be 
expanded by the binomial formula. Then it may be written 



8 Q = 2 V^gh [> - y*)* - (r> ~£ ly - ( —£ ^ - etc. 


By 


Each term of this expression is now integrable, and taking the 
limits of y as + r and — r the entire circle is covered, and Q is 
found. Finally, replacing r by J d there results 


Q = \ Trd 2 V 2 gh 



my s(d/ky ctc 

128 16384 


which is the theoretic discharge from the circular orifice. 

It is plain that this formula gives values which are always 
less than those found from the approximate formula of the first 
paragraph. Thus for h = d the quantity in the parenthesis 
is 0.992 and for h = 2 d it is 0.998. Hence the error in using the 
approximate formula is less than three-tenths of one peicent 
when the head on the center of the orifice is greater than twice 
its diameter. 

For most cases, then, the actual discharge from a circular 
vertical orifice of area a may be computed from 

q = c • a 2gh = 8.02 ca 'Vh ( 47 ) 1 

in which c is the coefficient of discharge. When h is smaller 
than two or three times the diameter of the orifice, and when pre ¬ 
cision is required, then 

q =[1 -0.007812 (d/h) 2 - 0.000306 (d/h) 4 ] 8.02 ca Vk ( 47 ) 2 

is the formula to be used. Here a may be taken from 1 able F 
(Art. 205 ) for the given diameter expressed in feet, h is to be 
taken in feet, and then q will be in cubic feet per second. 

































118 Chap. 5 . Flow of Water through Orifices 

Table 47 a gives values of c for circular orifices as determined 
by Hamilton Smith in a discussion of all the best experiments.* 
They apply only to standard orifices with definite inner edges. 


Table 47 a. Coefficients for Circular Vertical Orifices 


Head 

Diameter of Orifice in Feet 

n 

in Feet 

0.02 

0.04 

0.07 

0.1 

0.2 

0.6 

10 

0.4 


0.637 

0.624 

0.618 



' 

0.6 

0-655 

.630 

.618 

•613 

0.601 

0.593 


0.8 

.648 

.626 

.615 

.610 

.601 

•594 

0.590 

1.0 

.644 

.623 

.612 

.608 

.600 

•595 

•591 

i -5 

•6 37 

.618 

.608 

.605 

.600 

•596 

•593 

2.0 

.632 

.614 

.607 

.604 

•599 

•597 

•595 

2-5 

.629 

.612 

.605 

.603 

•599 

•598 

•596 

3 -o 

.627 

.611 

.604 

.603 

•599 

•598 

•597 

4.0 

.623 

.609 

.603 

.602 

•599 

•597 

•596 

6.0 

.618 

.607 

.602 

.600 

•598 

•597 

•596 

8.0 

.614 

•605 

.601 

.600 

•598 

•596 

•596 

10.0 

.611 

.603 

•599 

•598 

•597 

•596 

•595 

20.0 

.601 

•599 

•597 

•596 

•596 

•596 

•594 

50.0 

•596 

•595 

•594 

•594 

•594 

•594 

•593 

100.0 

•593 

•592 

•592 

•592 

•592 

•592 

•592 


The table shows that the coefficient of discharge decreases as 
the size of the orifice increases, and that in general it also de¬ 
creases as the head increases. In this table the coefficients 
found above the horizontal lines in the last three columns are 
to be used in the exact.formula ( 47 ) 2 and all others in the approxi¬ 
mate formula ( 47 )x. 

For example, let it be required to find the discharge through 
a standard circular orifice, 2 inches in diameter, under a head 
of 2.35 feet. First, 2 inches = 0.1667 feet, and by interpolation 
in Table 47 <z the coefficient c is found to be 0.602. Next, from 
Table F at the end of this book, the area a is 0.02182 square 
feet. Then formula ( 47 )x gives the discharge q as 0.161 cubic 
feet per second. As the coefficient is probably liable to an error 


* Hydraulics (London and New York, 1886), p. 59. 


























Circular Vertical Orifices. Art. 47 


119 


of one or two units in the last figure, the third figure of this value 
of q is subject to the same uncertainty. 

Judd and King* determined in 1906 the following values of 
the coefficient of discharge for circular vertical orifices : 

Orifice diameter, inches 0.75 1.00 1.50 2.00 2.50 

Coefficient of discharge 0.611 0.6097 0.6085 0.6083 0.5956 

The heads under which the observations were made ranged from 
5 to 90 feet and the results showed no appreciable change in the 
coefficient of discharge due to increased head. For example 
the following are part of the results found for a 2-inch orifice : 

Head in feet = 5.00 9.08 17.79 36.12 57.70 92.01 

Coefficient c — 0.6084 0.6083 0.6080 0.6082 0.6081 0.6080 

Biltonf in 1907 made a series of experiments on orifices, rang¬ 
ing from 0.025 t° °-75 inches in diameter and determined the 
following coefficients for varying heads. 

Table 47 b . Coefficients of Discharge for Small Orifices 


Head 

h 




Diameter 

of Orifice in Inches 




in 

Feet 

0.025 

0.05 

0.10 

0.20 

0.30 

0.40 

0.50 

0.60 

0-75 

0.50 

0.748 

0.722 

0.690 

0.673 

0.665 

0.652 

0.645 

0.644 

0.632 

1.00 

4* 

00 

.717 

.680 

• 6 S 9 

.647 

.636 

.630 

.627 

.618 

2.00 

.748 

.708 

.666 

.642 

.630 

.624 

.621 

.618 

.613 

4.00 

00 

.697 

.652 

.630 

.627 

.624 

.621 

.618 

.613 

6.00 

.748 

.688 

.647 

.630 

.627 

.624 

.621 

.618 

.613 

8.00 


.683 

•645 

.630 

.627 

.624 

.621 

.618 

.613 


Experiments made in 1908 by Strickland J on standard orifices 
1 and 2 inches in diameter gave results for the coefficient of dis¬ 
charge very closely represented by the formula 

i 2 

c = 0.5925 + 0.01 8/Jrd* 

* Engineering News, Sept. 27, 1906. 

f Proceedings Victorian Institute of Engineers, Australia, 1908. 
t Transactions Canadian Society of Civil Engineers, 1909, vol. 23, p. 198. 
























120 


Chap. 5 . Flow of Water through Orifices 


where h is in feet and d in inches. Applying this formula to an 
orifice 2 inches in diameter under a head of 19 feet, c is found to 
be 0.5951 while the experiments indicated a value of 0.5947. 

Prob. 47 . Compute the probable actual discharge from a circular orifice 
8 inches in diameter, under a head of 15 inches. 


Art. 48 . Square Vertical Orifices 


If the size of an orifice in the side of a vessel is small compared 
with the head, the theoretic velocity of the outflowing water may 
be taken as V2 gh, where h is the head on the center of the orifice. 
IA>r a rectangular orifice under this condition the theoretic dis- 

charge is Q = bdV^fh 

where b is the width and d the depth of the orifice. When b 
is equal to d , the rectangle becomes a square. 

To deduce a more exact formula, let 
hi be the head on the upper edge of the 
orifice and h 2 that on the lower edge. 
Consider an elementary strip of area b • By 
at a depth y below the water level. The 
velocity of flow through this elementary 
strip is V2 gy, and the theoretic discharge per second through it is 

BQ = bBy V2 gy 


K 


h. 




y 


- 50 - 

Fig. 48. 


Integrating this between the limits h 2 and h\ , there results 

Q = § b V2g(fc f - hj) 

which is the true theoretic discharge from the orifice. 

To ascertain the error caused by using the approximate for¬ 
mula, let h be the head on the center of the rectangle; then h 2 
= h + 2 d an d h\ = h — § d. Developing by the binomial formula 
the values of hf and h{\ the last formula becomes 


Q = bd~V 2 gh 



(d/hY 

96 



and this shows that the discharge computed by using the approx¬ 
imate formula is always too great. For h = d, the quantity in 














Square Vertical Orifices. Art. 48 


121 


the parenthesis is 0.989, and for h = 2d, it is 0.997. Accordingly, 
the error of the approximate formula is only three-tenths of 
one percent when the head on the center of the rectangle is 
twice the depth of the orifice. 

For most cases, then, the actual discharge from a square 
vertical orifice may be very approximately found from 

q = c • b 2 V2gh = 8.02 cb 2 Vh ( 48 )i 

where b is the side of the square and c is the coefficient of dis¬ 
charge. When h is smaller than two or three times the side of the 
orifice, and when precision is required, 

Q= 5-347 cb(Jh~ — h$) ( 48) 2 

is the formula to be used. The linear quantities are to be taken 
in feet, and then q will be in cubic feet per second. 

Table 48 gives values of the coefficient c for standard square 
orifices, taken from a more extended one formed by Hamilton 


Table 48 . Coefficients for Square Vertical Orifices 


Head 

h 

in Feet 



Side of the Square in Feet 


1 

0.02 

0.04 

0.07 

O.I 

0.2 

0.6 

1.0 

O O 

'0 L 

0.660 

0.643 

.636 

0.628 

.623 

0.621 

.617 

0.605 

0.598 


0.8 

.652 

.631 

.620 

.615 

.605 

.600 

0-597 

1.0 

.648 

.628 

.618 

.613 

.605 

.601 

•599 

i -5 

.641 

.622 

.614 

.610 

.605 

.602 

.601 

2.0 

.637 

.619 

.612 

.608 

.605 

.604 

.602 

2-5 

•634 

.617 

.610 

.607 

.605 

.604 

.602 

3 -° 

.632 

.616 

.609 

.607 

.605 

.604 

.603 

4.0 

.628 

.614 

.608 

.606 

.605 

.603 

.602 

6.0 

.623 

.612 

.607 

.605 

.604 

.603 

.602 

8.0 

.619 

.610 

.606 

.605 

.604 

.603 

.602 

10.0 

.616 

.608 

.605 

.604 

.603 

.602 

.601 

20.0 

.606 

.604 

.602 

.602 

.602 

.601 

.600 

50.0 

.602 

.601 

.601 

.600 

.600 

•599 

•599 

100.0 

•599 

•598 

.598 

.598 

•598 

•598 

•598 


Smith in 1886 by the discussion of all the best experiments. 
It is seen that the coefficient decreases as the size of the orifice 




























122 


Chap. 5 . Flow of Water through Orifices 


increases and as the head increases. Comparing this table 
with Table 47 a it is seen that the coefficient of discharge for a 
square is always slightly larger than that for a circle having a 
diameter equal to the side of the square. The values above the 
horizontal lines in the last three columns are to be used in the 
exact formula ( 48) 2 when precision is required, and all other values 
in the approximate formula ( 48 )i. 

There are few recorded experiments on large square orifices. 
Ellis measured the discharge from a vertical orifice 2 feet square* 
and deduced the following coefficients for use in the approximate 
formula : 

for h= 2.07 feet, c = 0.611 

for h = 3.05 feet, c = 0.597 

for h = 3.54 feet, c = 0.604 

which indicate that a mean value of 0.60 may be used for large 
square orifices under low heads. 

Prob. 48 . Find from the table the coefficient for an orifice 3 inches square 
when the head on its center is 1.8 feet. 

Art. 49 . Rectangular Vertical Orifices 

The theoretic formulas of Art. 48 apply to rectangles of width 
b and depth d , and the approximate formula for computing the 

actual discharge is _ 

q = cbd V 2 gh = 8.02 cbd V li (49) 

in which c is the coefficient of discharge, b the width and d the 
depth of the rectangular orifice, and h the head on its center. 

Table 49 gives values of the coefficient c which have been 
compiled and rearranged from the discussion given by Fanning.! 
It is seen that the variation of c with the head follows the 
same law as for circles and squares. It is also seen that for a 
rectangle of constant breadth the coefficient increases as the 
depth decreases, from which it is to be inferred that for a rec¬ 
tangle of constant depth the coefficient increases with the breadth, 

* Transactions American Society of Civil Engineers, 1876, vol. 5, p. 92. 
t Treatise on Water Supply Engineering (New York, 1888), p. 205. 


Velocity of Approach. Art. 50 123 


Table 49 . Coefficients for Rectangular Orifices 

i Foot Wide 


Head 

h 

in Feet 



Depth of Orifice in 

Feet 



0 125 

0 25 

0 50 

0 75 

I O 

15 

2 0 

0.4 

0.634 

0-633 

0.622 





0.6 

•633 

•633 

.619 

O.614 


• 


0.8 

•633 

•633 

.618 

.612 

0.608 



1.0 

.632 

.632 

.618 

.612 

.606 

O.626 


i -5 

.630 

.631 

.618 

,6ll 

.605 

.626 

0.628 

2.0 

.629 

.630 

.617 

.6ll 

.605 

.624 

.630 

2-5 

.628 

.628 

.616 

,6ll 

.605 

.616 

.627 

30 

.627 

.627 

.615 

.6lO 

.605 

.614 

.619 

4.0 

.624 

.624 

.614 

.609 

.605 

.612 

.6t6 

6.0 

.615 

.615 

.609 

.604 

.602 

.606 

.610 

8.0 

.609 

.607 

.603 

.602 

.601 

.602 

.604 

10.0 

.606 

.603 

.601 

.601 

.601 

.601 

.602 

20.0 




'6oi 

.601 

.601 

.602 


and this is confirmed by other experiments. The value of c for a 
rectangular orifice is seen to be only slightly larger than that for 
a square whose side is equal to the depth of the rectangle. All 
the coefficients in this table are for the above approximate for¬ 
mula, since that formula was used in computing them. 

A comparison of the values of c for the orifice one foot square 
with those in the last article shows that the two sets of coefficients 
disagree, these being about one percent greater. This is prob¬ 
ably due to the less precise character and smaller number of 
experiments from which they were deduced. 

Prob. 49 . What constant head is required to discharge 5 cubic feet 
of water per second through an orifice 3 inches deep and 12 inches long? 


Art. 50 . Velocity of Approach 

It was shown in Art. 24 that the theoretic velocity of flow from 
an orifice is greater then V2 gh when the ratio of the cross-sec¬ 
tion of the orifice to that of the vessel or tank is not small. The 
same is true for the actual velocity, but formula ( 24 ) x must be 


























124 


Chap. 5 . Flow of Water through Orifices 


modified because it takes no account of the contraction of the 

jet. Let v be the velocity at the contracted section of the jet 

and a' the area of that section; let v x be the velocity through the 

horizontal cross-section A of the vessel; then a'v = Av v But 

if a be the area of the orifice and c' the coefficient of contraction, 

then a' equals ac' and hence c'av = Ai\. Now the effective 

head on the orifice is 2 

H = /f+ — 

2 g 


and the velocity v is given by c l V2 gH where c 1 is the coefficient 
of velocity. Substituting in the last equation v 2 /2gc 2 for H and 
cva/A for v 1} and noting that Cyc' is equal to the coefficient of 
discharge c, it reduces to _ 


V = Cy 


c 2 (a/A ) 2 



which is the velocity of the jet at a section distant from the orifice 
about one-half its diameter. The discharge q is found by multi¬ 
plying this by the area c'a of that cross-section, whence 


q = ca 



2gh 

c 2 (a/A ) 2 


2 gh 


(i/c) 2 -(a/A)‘ 



is the formula for the actual discharge, and this includes no 
coefficient except that of discharge. 


These formulas apply to orifices of any kind, and when c 
equals unity, they reduce to the theoretic expressions established 
in Art. 24 . When a/A is less than 1/5, as is almost always the 
case in practice, the last formula may be written, with sufficient 
precision, 


q = (1 + h (« ca/A) 2 )ca v 2 gh 


( 50 ), 


For example, let a square tank, 4X4 feet in horizontal cross-sec¬ 
tion, have a standard square orifice one square foot in area, and 
let the head on its center be 16 feet. From Table 48 the coeffi¬ 
cient of discharge is 0.60, and the formula gives 

q = (1 + 0.0007) X 0.60 X 1 X 8.02 X 4 = 19.3 cubic feet per second 

For this case it is seen that the influence of velocity of approach 
is expressed by the addition of 0.0007 to unity, which is an in- 











Velocity of Approach. Art. 50 


125 


crease of less than one-tenth of one percent. In general the 
increase in discharge due to velocity of approach is expressed, 
when a/A is not greater than 1/5, by | c*a(a/ T) 2 V 2gh. 

A common case is that where the vessel or tank is of large 
horizontal and small vertical cross-section, and where the water 
approaches the orifice with a horizontal velocity, as in a canal 
or conduit. Here let A be the area of the vertical cross-section 
of the vessel, a the area of the orifice, and h the head on its center. 
Then, if the head h be large compared with the depth of the orifice, 
the same reasoning applies as in Art. 24 , the theoretic velocity 
is given by ( 24 ) x and the actual discharge by ( 50 ) 2 . 

When the head h is not large, let hi and h 2 be the heads on the 
upper and lower edges of the orifice, which is taken as rectangular 
and of the width b. Let v be the 
velocity of approach, which is re¬ 
garded as uniform over the area A . 

Then by the same reasoning as 
that in Art. 24 , the theoretic ve¬ 
locity in the plane of the orifice 
at the depth y below the water 
level is given by V 2 = 2gy -f v 2 . 

The theoretic discharge through an elementary strip of the 
length b and the depth 8 y now is 

SQ — (2 gy + v 2 )' 1 b 8 y 

and, by integration between the limits h 2 and hi, the total theoretic 
discharge is found. If v 2 /2g be replaced by h 0 , the head which 
would cause the velocity v, the theoretic discharge is 

Q = 15 Vig [{h 2 + ho ) 2 — {hi + ho) 1 ] ( 50 ) t 

and the actual discharge q is found by multiplying this by a 
coefficient of discharge. When there is no velocity of approach, 
the formula reduces to that found in Art. 49 for this case. 

Prob. 50 a. When n is a small quantity compared with unity, show 
that (1 + rift = 1+2 n, and that 1/(1 + w) - 1 ~ n - Deduce formula 
( 50) 3 from ( 50 ) 2 . 



































126 


Chap. 5 . Flow of Water through Orifices 


Prob. 50 b. In the case of horizontal approach, as seen in Fig. 50 , com¬ 
pute the discharge when b — 4 feet, h . 2 = 0.8 feet, h x = o, v = 2. 5 feet per 
second, and c = 0.6. 

Art. 51 . Submerged Orifices 

It is shown in Art. 23 that the effective head li which causes 
the flow from a submerged orifice is the difference in level be¬ 
tween the two water surfaces. The discharge from such an orifice, 
its inner edge being a sharp definite one, as in Fig. 43 a, has been 
found by experiment to be slightly less than when the flow oc¬ 
curs freely into the air, and hence the values of the coefficients 
of discharge are slightly smaller than those given in Tables 47 a, 
475 , 48 , 49 . For large orifices and large heads the difference is very 
small, and for orifices one inch square under six inches head it is 
about 2 percent. In all cases of submerged orifices the discharge 
is to be found from q = ru V2 gh. 

Table 51 gives values of the coefficient of discharge for sub¬ 
merged orifices as determined from experiments made by Hamil¬ 
ton Smith in 1884. The depth of submergence of the orifices 
varied from 0.57 to 0.73 foot. As a mean value of the coefficient 
of discharge for standard submerged orifices 0.6 is frequently 
used. 

Table 51 . Coefficients for Submerged Orifices 


Effective 
Head in 
Feet 

Size of Orifice in Feet 

Circle 

0.05 

Square 

0.05 

Circle 

0.1 

Square 

0.1 

Rectangle 

0.05 X 0.3 

o -5 

0.615 

0.619 

0.603 

0.608 

0.623 

1.0 

.610 

.614 

.602 

.606 

.622 

i -5 

.607 

.612 

.600 

.605 

.621 

2.0 

.605 

.610 

•599 

.604 

.620 

2.5 

.603 

.608 

•598 

.604 

.619 

3 -o 

.602 

.607 

•598 

.604 

.618 

4.0 

.601 

.606 

•598 

.604 



The theoretic discharge from a submerged orifice is the same 
for the same effective head //, whatever be its distance below water 
level. The theoretic velocity in all parts of the orifice is the 


















Suppression of the Contraction. Art. 52 


127 


-—j- 

B 


D F O 

Fig. 51 . 


E 


same, as may be proved from Fig. 51 , where the triangles ACD 
and BCE represent the distribution of pressure on AC and BC 
when the orifice is closed (Art. 17 ). Mak¬ 
ing CF equal to CE and drawing BE, the 
unit-pressure on BC is seen to have the 
constant value DF. Now when the orifice 
is opened, the velocity at any point de¬ 
pends on the unit-pressure there acting, 
as seen by ( 23 )i, and accordingly the the¬ 
oretic velocity is uniform over the section. 

For this reason the coefficients of discharge probably vary less 
with the head than for the previous cases. 

Submerged orifices are used for canal-locks, tide-gates, filter- 
beds, for the discharge of waste water through dams, and for the 
admission of water from a canal to a power-plant. The inner 
edges of such orifices are usually rounded, and the coefficient of 
discharge may then be higher than 0.9 (Art. 53 ). 

Prob. 51 . An orifice one inch square in a gate, such as shown in Fig. 19 < 7 , 
is 4.1 feet below the higher water level and 3.1 feet below the lower level. 
Compute the discharge in cubic feet per second, and also in gallons per minute. 


Art. 52 . Suppression of the Contraction 

When a vertical orifice has its lower edge at the bottom of 
the reservoir, as shown at A in Fig. 52 , the particles of water 

flowing through its lower portion move in 
lines nearly perpendicular to the plane of the 
orifice, or the contraction of the jet does not 
form on the lower side. This is called a case 
of suppressed or incomplete contraction. The 
Fi s- 52 - same thing occurs, but in a lesser degree, when 

the lower edge of the orifice is near the bottom, as shown at B. 
In like manner, if an orifice be placed so that one of its vertical 
edges is at or near a side of the reservoir, as at C, the contrac¬ 
tion of the jet is suppressed upon one side, and if it be placed 
at the lower corner of the reservoir suppression occurs both upon 
one side and the lower part of the jet. 





























128 Chap. 5 . Flow of Water through Orifices 

The effect of suppressing the contraction is, of course, to in¬ 
crease the cross-section of the jet at the place where full contrac¬ 
tion would otherwise occur, and it is found by experiment 
that the discharge is likewise increased. Experiments also show 
that more or less suppression of the contraction will occur unless 
each edge of the orifice is at a distance at least equal to three times 
its least diameter from the sides or bottom of the reservoir. 

The experiments of Lesbros and Bidone furnish the means 
of estimating the increased discharge caused by suppression of the 
contraction. They indicate that for square orifices with con¬ 
traction suppressed on one side the coefficient of discharge is 
increased about 3.5 percent, and with contraction suppressed 
on two sides about 7.5 percent. For a rectangular orifice 
with the contraction suppressed on the bottom edge the per¬ 
centages are larger, being about 6 or 7 percent when the length 
of the rectangle is four times its height, and from 8 to 12 percent 
when the length is twenty times the height. The percentage 
of increase, moreover, varies with the head, the lowest heads 
giving the lowest percentages. 

It is apparent that suppression of the contraction should be avoided 
if accurate results are desired. The experiments from which the above 
conclusions are deduced were made upon small orifices with heads 
less than 6 feet, and it is not known how they will apply to large ori¬ 
fices under high heads. For a rectangular orifice of length about 
three times its height, with contraction suppressed on the ends and 
bottom, the coefficient of discharge is probably about 0.75. 

Prob. 52 . Compute the probable discharge from a vertical orifice one 
foot square when the head on its upper edge is 4 feet, the contraction being 
suppressed on the lower edge. Compute the discharge for the same data 
when contraction is suppressed on all sides. 

Art. 53 . Orifices with Rounded Edges 

When the inner edge of the orifice is made rounded, as shown 
in Fig. 53 , the contraction of the jet is modified, and the dis¬ 
charge is increased. With a slight degree of rounding, as at 
A , a partial contraction occurs; but with a more complete round¬ 
ing, as at C, the particles of water issue perpendicular to the plane 


Water Measurement by Orifices. Art. 54 


129 


of the orifice and there is no contrac¬ 
tion of the jet. If a be the area of 
the least cross-section of the orifice, 
and a' that of the jet, the coefficient 
of contraction as defined in Art. 44 is 

c r = a /a ( 53 ) 




For a standard orifice with sharp inner Fi s- 53 

edges (Art. 43 ) the mean value of c' is 0.62, but for an orifice 
with rounded edges c' may have any value between 0.62 and 1.0, 
depending upon the degree of rounding. 


The coefficient of discharge c for standard orifices has a mean 
value of about 0.61; this is increased with rounded edges and may 
have any value between 0.61 and 1.0. A rounded interior edge 
in an orifice is therefore always a source of error when the object 
of the orifice is the measurement of the discharge. If a contract 
provides that water shall be gaged by standard orifices, care should 
always be taken that the interior edges do not become rounded 
either by accident or by design. 


Prob. 53 . When an orifice with rounded edges has a coefficient of 
velocity of 0.88 and a coefficient of discharge of 0.75, find the coefficient of 
contraction of the jet. 


Art. 54 . Water Measurement by Orifices 

In order that water may be accurately measured by the use 
of orifices many precautions must be taken, some of which have 
already been noted, but may here be briefly recapitulated. The 
area of the orifice should be small compared with the size of the 
reservoir in order that velocity of approach may not exist, or 
if this cannot be avoided, it should be taken into account by for¬ 
mula ( 50 )i. The inner edge of the orifice must have a definite 
right-angled corner, and its dimensions are to be accurately 
determined. If the orifice be in wood, care should be taken that 
the inner surface be smooth, and that it be kept free from the 
slime which often accompanies the flow of water, even when appar¬ 
ently clear. That no suppression of the contraction may occur, 














130 


Chap. 5 . Flow of Water through Orifices 


the edges of the orifice should not be nearer than three times its 
least dimension to a side of the reservoir. 

Orifices under very low heads should be avoided, because 
slight variations in the head produce relatively large errors, and 
also because the coefficients of discharge vary more rapidly 
and are probably not so well determined as for cases where the 
head is greater than four times the depth. If the head be very 
low on an orifice, vortices will form which render any estimation 
of the discharge unreliable. 

The measurement of the head, if required with precision, 
must be made with the hook gage described in Art. 35 . For 
heads greater than two or three feet the readings of an ordinary 
glass gage placed upon the outside of the reservoir will usually 
prove sufficient, as this can be read to hundredths of a foot with 
accuracy. An error of o.oi foot when the head is 3.00 feet 
produces an error in the computed discharge of less than two- 
tenths of one per cent; for, the discharges being proportional 
to the square roots of the heads, the square root of 3.01 divided 
by the square root of 3.00 equals 1.0017. For the rude measure¬ 
ments in connection with the miner's inch a common foot-rule 
will usually suffice. 

The effect of temperature upon the discharge remains to 
be noticed; this is only appreciable with small orifices and under 
low heads and hence such orifices and heads are not desirable 
in precise measurements. Unwin found that the discharge was 
diminished one percent by a rise of 144 0 in temperature; his 
orifice was a circle 0.033 feet ' m diameter under heads ranging 
from 1.0 to 1.5 feet. Hamilton Smith found that the discharge 
was diminished one percent by a rise of 55 0 in temperature; his 
orifice was a circle 0.02 feet in diameter under heads ranging 
from 0.56 to 3.2 feet. 

The coefficients given in the tables of this chapter may be supposed 
liable to a probable error of about two units in the third decimal place : 
thus a coefficient 0.615 should really be written 0.615 T °-oo2 ; that is, 
the actual value is as likely to be between 0.613 an d 0.617 as to be 
outside of those limits. The probable error in computed discharges 


The Miner’s Inch. Art. 55 


131 


due to the coefficient is hence nearly one-half of one percent. To 
this are added the errors due to inaccuracy of observation, so that it 
is thought that the probable error of careful work with standard cir¬ 
cular orifices is at least one percent. The computed discharges are 
hence liable to error in the third significant figure, so that it is useless 
to carry numerical results beyond three figures when based upon tabu¬ 
lar coefficients. As a precise method of measuring small quantities 
of water, standard orifices take a high rank when the observations are 
conducted with care. 

Prob. 54 . If e is a small error in measuring the head h, show that the 
error in the computed discharge q due to this cause is qc/2/1. 

Art. 55 . The Miner ’s Inch 

The miner's inch may be roughly defined to be the quantity 
of water which will flow from a vertical standard orifice one inch 
square, when the head on the center of the orifice is 6| inches. 
From Table 48 the coefficient of discharge is seen to be about 
0.623 an d accordingly the actual discharge from the orifice in 
cubic feet per second is q = yjq X 0.623 X 8.02 V6.5/12 = 0.0255 
and the discharge in one minute is 60X0.255 — 1.53 cubic 
feet. The mean value of one miner’s inch is therefore about 
1.5 cubic feet per minute. 

The actual value of the miner’s inch, however, differs con¬ 
siderably in different localities. Bowie states that in different 
counties of California it ranges from 1.20 to 1.76 cubic feet per 
minute.* The reason for these variations is due to the fact that 
when water is bought for mining or irrigating purposes, a much 
larger quantity than one miner’s inch is required, and hence larger 
orifices than one square inch are needed. Thus at Smartsville 
a vertical orifice or module 4 inches deep and 250 inches long, with 
a head of 7 inches above the top edge, is said to furnish 1000 
miner’s inches. Again, at Columbia Hill, a module 12 inches 
deep and 124 inches wide, with a head of 6 inches above the upper 
edge, is said to furnish 200 miner’s inches. In Montana the cus¬ 
tomary method of measurement is through a vertical rectangle, 


* Treatise on Hydraulic Mining (New York, 1885), p. 268. 



132 Chap 5 . Flow of Water through Orifices 

i inch deep, with a head on the center of the orifice of 4 inches, 
and the number of miner’s inches is said to be the same as the 
number of linear inches in the rectangle; thus under the given 
head an orifice 1 inch deep and 60 inches long would furnish 
60 miner’s inches. The discharge of this is said to be about 
1.25 cubic feet per minute, or 75 cubic feet per hour. 

The following are the values of the miner’s inch in different 
parts of the United States; in California and Montana it is es¬ 
tablished by law that 40 miner’s inches shall be the equivalent 
of one cubic foot per second, and in Colorado 38.4 miner’s inches 
is the equivalent. In other States and Territories there is no 
legal value, but by common agreement 50 miner’s inches is 
the equivalent of one cubic foot per second in Arizona, Idaho, 
Nevada, and Utah; this makes the miner’s inch equal to 1.2 
cubic feet per minute. 

A module is an orifice which is used in selling water, and which 
under a constant head is to furnish a given number of miner’s 
inches, or a given quantity per second. The size and proportions 
of modules vary greatly in different localities, but in all cases the 
important feature to be observed is that the head should be main¬ 
tained nearly constant in order that the consumer may receive 
the amount of water for which he bargains, and no more. 

The simplest method of maintaining a constant head is by 
placing the module in a chamber which is provided with a gate 
that regulates the entrance of water from the main reservoir 
or canal. This gate is raised or lowered by an inspector once 
or twice a day so as to keep the surface of the water in the 
chamber at a given mark. This plan is a costly one, on account 
of the wages of the inspector, except in works where many modules 
are used and where a daily inspection is necessary in any event, 
and it is not well adapted to cases where there are frequent and 
considerable fluctuations in the water surface of the feeding canal. 

Numerous methods have been devised to secure a constant head 
by automatic appliances; for instance, the gate which admits water 
into the chamber may be made to rise and fall by means of a float 
upon the surface; the module itself may be made to decrease in size 


Loss of Energy or Head. Art. 56 133 

when the water rises, and to increase when it falls, by a gate or by 
a tapering plug which moves in and out and whose motion is con¬ 
trolled by a float. In another variety the head on an orifice is kept 
constant by placing it in the side of a vessel which is movable and 
whose vertical movement is proportional to the rise or fall of the water 
in the feeding channel or reservoir. These self-acting contrivances, 
however, are liable to get out of order, and require to be inspected 
more or less frequently.* Another method is to have the water flow 
over the crest of a weir as soon as it reaches a certain height.! 

The use of the miner’s inch, or of a module, as a standard for sell¬ 
ing water, is awkward and confusing, and for the sake of uniformity 
it is greatly to be desired that water should always be bought and sold 
by the cubic foot per second. Only in this way can comparisons readily 
be made, and the consumer be sure of obtaining exact value for his 
money. 

Prob. 55 . When a miner’s inch is 1.57 cubic feet per minute, how many 
miner’s inches will be furnished by a module 2 inches deep and 50 inches long 
with a head of 6 inches above the upper edge ? 

Art. 56 . Loss of Energy or Head 

A jet of water flowing from an orifice possesses by virtue of 
its velocity a certain kinetic energy, which is always less than the 
theoretic potential energy due to the head (Art. 26 ). Let h be 
the head and W the weight of water discharged per second, then 
the theoretic energy per second or the power of the jet, is 

K = Wh 

Let v be the actual velocity of the water at the contracted section 
of the jet; then the actual energy per second of the water as it 
passes that section is ^ _ jy . v 2 / 2 g 

Now let Ci be the coefficient of velocity (Art. 45 ); then 

v 2 = Ci • 2 gh 

and accordingly the actual energy of the jet per second is 

k = C\Wh 


* For descriptions of several see Engineering News. Dec. 17, 1908. 
f Foote, Transactions American Society Civil Engineers, 1837, vol. 16, p. 134* 


134 


Chap. 5 . Flow of Water through Orifices 


The efficiency of the jet, or the ratio of the actual to the theoretic 


energy, now is 


e = k/K = ci 2 


( 56 ) 


which is a number always less than unity. 

For the standard orifice the mean value of c Y is 0.98, and hence 
a mean value of c 2 is 0.96. The actual energy of a jet from such 
an orifice is hence about 96 percent of the theoretic energy, and 
the loss of energy is about 4 percent. This loss is due to the 
frictional resistance of the edges of the orifice, whereby the energy 
of pressure or velocity is changed into heat. 

In the plane of the standard orifice the velocity is slower than 
at the contracted section since the area there is greater. If 
i\ be this velocity, a the area of the orifice, and a that of the jet 
at the contracted section, it is clear that ai\ = a'v or 1\ = c v , 
where c' is the coefficient of contraction 0.62. The kinetic energy 
in the planeof the orifice is W • v 2 /2g,or 0.37 Wv 2 /2 g , or 0.37 Wh. 
Thus, in the plane of the orifice 4 percent of the theoretic energy 
is lost overcoming friction, 37 percent is in the form of kinetic 
energy, and the remaining 59 percent exists in the form of 
pressure energy. This 59 percent is transformed into kinetic 
energy when the water has reached the contracted section. 

In hydraulics the terms “energy” and “head” are often used 
as synonymous, although really energy is proportional to head. 
Thus the pressure-head that causes the flow is h and the velocity- 
head of the issuing jet is v 2 /2g, and these are proportional to the 
theoretic and effective energies. The lost head W is the differ¬ 
ence of these, or *> 



and this applies not only to an orifice but to any tube or pipe. 
Inserting for t 2 its value, this becomes 


h' = (1 — C1 2 ) h 


which gives the lost head in terms of the total head. Inserting 
for h its value in terms of v reduces this to 



Discharge under a Dropping Head. Art. 57 


135 


which gives the lost head in terms of the velocity-head. Thus, 
for an orifice whose coefficient of velocity is 0.97 the lost head 
h' is 0.060 h or 0.063 2 £- For the standard orifice the lost 

head h! is 0.040 h or 0.041 v 2 /2g. For the standard orifice h f can 
also be expressed as o.n v x 2 /2g, where v x is the velocity in the 
plane of the orifice. 

Prob. 56 . What is the loss of head in an orifice whose coefficient of 
velocity is unity ? 


Art. 57 . Discharge under a Dropping Head 


If a vessel or reservoir receives no inflow of water while an 
orifice is open, the head drops and the discharge decreases in 
each successive second. Let H be the head on the orifice at a 
certain instant, and h the head t seconds later; let A be the area 
of the uniform horizontal cross-section of the vessel, and a the 
area of the orifice. Then, the theoretic time t is given by the 
second formula in Art. 32 . To determine the actual time 
the coefficient of discharge must be introduced. Referring to 
the demonstration, it is seen that a'\/2gyU is the theoretic dis¬ 
charge in the time Bt ; hence the actual discharge is c • a V 2gyBt, 
and accordingly a in the above-mentioned formula is to be re¬ 
placed by ca, or 

< = -^=(Vff-VA) (57) 

ca V2 g 


is the practical formula for the time in which the water level 
drops from H to h. In using this formula c is to be taken from 
the tables of this chapter, an average value being selected cor¬ 
responding to the average head. 

Experiments have been made to determine the value of c by 
the help of this formula; the liquid being allowed to flow, A, 
a , H, h, and t being observed, whence c is computed. In this 
way c for mercury has been found to be about 0.62.* Only ap¬ 
proximate mean values can be found in this manner, since c 
varies with the head, particularly for small orifices (Art. 47 ). 
For a large orifice the time of descent is usually so small that it 


* Downing’s Elements of Practical Hydraulics (London, 1875), p. 187. 




136 Chap. 5 . Flow of Water through Orifices 

cannot be noted with precision, and the friction of the liquid on 
the sides of the vessel may also introduce an element of uncer¬ 
tainty. ^Further, when h is small, a vortex forms which renders 
the formula unreliable. This experiment has therefore little 
value except as illustrating and confirming the truth of the theo¬ 
retic formulas. 

The discharge in one second when the head is H at the be¬ 
ginning of that second is found as follows: the above equation 
may be written in the form 

V# - tea ^2g/2A = Vh 

By squaring both members, transposing, and multiplying by 
A, this may be reduced to 

A (H — h) = tea V 2g (V H — tea V2^/4 A ) 

But the first member of this equation is the quantity discharged 
in t seconds; therefore the discharge in the first second is 

q = ca V2g( V /7 — ca V 2g/4 A) 

If A =00, this becomes ca ^2gh, which should be the case, for 
then H would remain constant. At the end of the first second 
the water level has fallen the amount q/A, so that the head at 
the beginning of the second second is H — q/A. 

For example, let an orifice one foot square in a reservoir 
of 10 square feet section be under a head of 9 feet, and c = 0.602. 
Then the discharge in one second is 13.9 cubic feet, and the head 
drops to 7.61 feet. The discharge in the next second is 12.7 
cubic feet, and the head drops to 6.34 feet. 

Prob. 57 . Find the time required to discharge 480 gallons of water from 
an orifice 2 inches in diameter at 8 feet below the water level when the cross- 
section of the tank is 4 X 4 feet. 

Art. 58 . Emptying and Filling a Canal Lock 

A canal lock is emptied by opening one or more orifices in the 
lower gates. Let a be their area and Ii the head of water on 
them when the lock is full; let A be the area of the horizontal 
cross-section of the lock. Then in the first formula of the last 


Emptying and Filling a Canal Lock. Art. 58 137 


article h — o, and the time of emptying the lock is 

t = 2A 'VH/ca V2g ( 58 ) 

If the discharge be free into the air, H is the distance from the 
center of the orifice to the level of the water in the lock when 
filled; but if, as is usually the case, the orifices be below the level 
of the water in the tail bay, H is the difference in height between 
the two water levels. The tail bay is regarded as so large com¬ 
pared with the lock that its water level remains constant during 
the time of emptying. 

For example, let it be required to find the time of emptying 
a canal lock 8o feet long and 20 feet wide through two orifices 
each of 4 square feet area, the head upon which is 16 feet when the 
lock is filled. Using for c the value 0.6 for orifices with square 
inner edges, the formula gives 


/ = 


2 X 80 X 20 X 4 
0.6 X 8 X 8.02 


= 333 seconds = 5! minutes 


If, however, the circumstances be such that c is 0.8, the time is 
about 250 seconds, or 4^ minutes. It is therefore seen that it 
is important to arrange the orifices of discharge in canal locks 
with rounded inner edges. 


H 


V 

I 

L 


The filling of the lock is the reverse operation. Here the water 
in the head bay remains at a constant level, and the discharge 
through the orifices in the 
upper gates decreases with : --f. t~* * 

the rising head in the lock. 

Let H be the effective head 
on the orifices when the 
lock is empty, and y the 
effective head at any time 
/ after the beginning of the 
discharge. The area of the 
section of the lock being 




Head Bay i ___ 

^ Lock r 

I 11 


_ Ta il Ba y 


Fig. 58. 


A, the quantity A By is discharged in the time St, and this is equal 
to ca v/2 gy St, if a be the area of the orifices and c the coefficient 
of discharge. Hence the same expression as ( 58 ) results, and the 


























138 Chap. 5 . Flow of Water through Orifices 

times of filling and emptying a lock are equal if the orifices are 
of the same dimensions and under the same heads. The area 
required for the orifices may be found for any case from ( 58 ) 
when A, H , /, and c are given. 

Prob. 58. A lock 90 feet long and 20 feet wide, with a lift of 12 feet, 
contains a boat weighing 500 net tons. When the lock is emptied in order 
to lower the boat, how much water flows from the lower orifices? If the 
cross-section of these orifices is 12.3 square feet and c — 0.7, what is the time 
of emptying? 

Art. 59 . Computations in Metric Measures 

Most of the formulas of this chapter are rational and may be used 
in all systems of measures. The coefficients of contraction, velocity, 
and discharge are abstract numbers, which are the same in all sys¬ 
tems, like the constants of mathematics. In the metric system the 
area a is to be taken in square meters, the head h in meters, V 2 g 
as 4.427, and then the discharge q will be in cubic meters per second. 

(Art. 47) For standard circular vertical orifices the formulas 
(47)! and (47) 2 apply to the metric system if 8.02 be replaced by 4.427. 
In using these the coefficient c may be taken from Table 59a which 
has been adapted to metric arguments from Table 47. For example, if 


Table 59a. Coefficients for Circular Vertical Orifices 

Arguments in Metric Measures 


Head 

L 

Diameter of Orifice in Centimeters 

in Meters 

1 

2 

3 

6 

18 

30 

O.I 

0.642 

0.626 

0.619 




0.2 

•639 

.619 

.613 

0.601 

o-593 


0-3 

•634 

.613 

.608 

.600 

•595 

0.591 

o-5 

.626 

.609 

.605 

.600 

.596 

•593 « 

0.7 

.620 

.607 

.603 

•599 

.598 

'596 

I. 

.619 

.605 

.602 

•599 

•598 

•597 

i-5 

.614 

.604 

.601 

•598 

•597 

•596 

2. 

.611 

.603 

.600 

•597 

•596 

•596 

3- 

.607 

.600 

.598 

•597 

.596 

•595 

6. 

.600 

•597 

•596 

•596 

•595 

•594 

IS- 

•596 

•595 

•594 

•594 

•594 

•593 

30- 

•593 

•592 

•592 

•592 

•592 

•592 
































Computations in Metric Measures. Art. 59 


139 


the diameter of the orifice is 2.5 centimeters and the head on its center 
is 0.6 meters, interpolation in the table gives the value of c as 0.606. 

(Art. 48 ) For standard square vertical orifices the formulas 
( 48 )! and ( 48) 2 are changed to the metric system by substituting 4.427 
for 8.02 and 2.951 for 5.347. Table 59 b gives values of the coefficient 
c for arguments in metric measures. 

Table 59 b. Coefficients for Square Vertical Orifices 


Arguments in Metric Measures 


Head 

h 

in Meters 


Side of the Square in Centimeters 


1 

2 

3 

6 

12 

. 

30 

O.I 

0.2 

0.652 

.648 

0.632 

.624 

0.622 

.617 

0.605 

0.598 


0-3 

.636 

.619 

•613 

.605 

.601 

0-599 

0-5 

.628 

.618 

.610 

.605 

.602 

.601 

0.7 

.625 

.612 

.607 

.605 

.604 

.602 

1.0 

.620 

.610 

.607 

.605 

.604 

.603 

i -5 

.618 

.609 

.606 

.604 

.603 

.602 

2. 

.614 

.608 

.605 

.604 

•603 

.602 

3 - 

.611 

.606 

.604 

.603 

.602 

.601 

6. 

.605 

.603 

.602 

.602 

.601 

.600 

I 5 - 

.601 

.601 

.600 

.600 

•599 

•599 

30 - 

•598 

•598 

•598 

•598 

598 

•598 


(Art. 49 ) Table 49 has not been transformed into one with metric 
arguments, as it applies only to the special case where the rectangular 
orifice is one foot wide. If the heads in the first column are changed 
into meters, by writing 0.12 meters for 0.4 feet, 0.18 meters for 0.6 
feet, etc., and the numbers at the top are changed into centimeters 
by writing 3.8 centimeters for 0.125 feet, 7.6 centimeters for 0.25 feet, 
etc., the table will be ready for use with metric arguments for rec¬ 
tangular orifices 30.5 centimeters wide. 

(Art. 55 ) The miner’s inch, when the head on the center of the 
orifice is 16.5 centimeters, is 0.0433 cubic meters or 43.3 liters per 
minute. 

(Art. 58 ) In using ( 58 ) in the metric system, a and A are to be 
taken in square meters, H in meters, g as 9.80 meters per second 
per second, and V2 g as 4.427 ; q will then be found in cubic meters. 

Prob. 59 u. Michelotti found the range of a jet to be 6.25 meters on 























140 


Chap. 5 . Flow of Water through Orifices 


a horizontal plane 1.41 meters below the vertical orifice, which was under a 
head of 7.19 meters. Compute the coefficient of velocity. 

Prob. 0%. An orifice 3 centimeters square was under a constant head 
of 4 meters, and during 230 seconds the jet flowed into a tank which was found 
to contain 1122 liters. Show that the coefficient of discharge was 0.612. 

Prob. 59 c. Find from the table the coefficient of discharge for a standard 
circular orifice 2.5 centimeters in diameter under a head of 2.5 meters. 

Prob. 59 d. Compute the discharge through a standard orifice 7.5 
centimeters square under a head of 8 meters. 

Prob. 59 c. Compute the time required to empty a canal lock 7 meters 
wide and 32 meters long through an orifice of 0.9 square meters area, the head 
on the center of the orifice being 5.1 meters when the lock is filled. 


Standard Weirs. Art. 60 


141 


CHAPTER 6 

FLOW OF WATER OVER WEIRS 
Art. 60 . Standard Weirs 

A weir is a notch in the top of the vertical side of a vessel or 
reservoir through which water flows. The notch is generally 
rectangular, and the word “weir” will be used to designate a 
rectangular notch unless otherwise specified, the lower edge of 
the rectangle being truly horizontal, and its sides vertical. The 
lower edge of the rectangle is called the “crest” of the weir. In 




Fig. 60 d is shown the outline of the most usual form, where the 
vertical edges of the notch are sufficiently removed from the 
sides of the reservoir or feeding canal, so that the sides of the 
stream may be fully contracted; this is called a weir with end 
contractions. In the form of Fig. 606 the edges of the notch are 
coincident with the sides of the feeding canal, so that the filaments 
of water along the sides pass over without being deflected from 
the vertical planes in which they move; this is called a weir with¬ 
out end contractions, or with end contractions suppressed. Both 








142 


Chap. 6 . Flow of Water over Weirs 


kinds of weirs are extensively used for the measurement of water 
in engineering operations. 



It is necessary in order to make accurate measurements of 
discharge by a weir that the same precaution should be taken as 
for orifices (Art. 54 ), namely, that the inner edge of the notch 

shall be a definite angular corner so that the 
water in flowing out may touch the crest 
only in a line, thus insuring complete con¬ 
traction, as in Fig. 61 . In precise observa¬ 
tions a thin metal plate will be used for a 
Fig. 60c. crest, while in common work it may be 

sufficient to have the crest formed by a plank of smooth hard 
wood with its inner corner cut to a sharp right angle and its outer 
edge beveled. The vertical edges of the weir should be made in 
the same manner for weirs with end contractions, while for those 
without end contractions the sides of the feeding canal should be 
smooth and be prolonged a slight distance beyond the crest. It 
is also necessary to observe the same precautions as for orifices 
to prevent the suppression of the contraction (Art. 52 ), namely, 
that the distance from the crest of the weir to the bottom of the 
feeding canal, or reservoir, should be greater than three times the 
head of water on the crest. For a weir with end contractions a 
similar distance should exist between the vertical edges of the 
weir and the sides of the feeding canal. A standard weir is one 
in which these arrangements have been carefully carried out. 


The head of water II upon the crest of a weir is usually much 
less than the breadth of the crest b. The value of II should not 
be less than o.i feet, and it should not exceed 4.5 feet in order 
to keep within the range of experiments on the standard weir. 
The least value of b in practice is about 0.5 feet, and it does not 
often exceed 20 feet. Weirs are extensively used for measuring 
the discharge of small streams, and for determining the quantity 
of water supplied to hydraulic motors; the practical importance 
of the subject is so great that numerous experiments have been 
made to ascertain the laws of flow, and the coefficients of discharge. 

Since the head on the crest of a weir is small, it must be deter- 







Standard Weirs. Art. 60 


143 


mined with precision in order to avoid error in the computed dis¬ 
charge. The hook gage illustrated in Art. 35 is generally used 
for accurate work in connection with hydraulic motors, and the 
simpler form, consisting of a hook set into a leveling rod, is usually 
of sufficient precision for many cases. For rough gagings of 
streams the heads may be determined by setting a post a few feet 
upstream from the weir and on the same level as the crest, and 
measuring the depth of the water over the top of the post by a 
scale graduated to tenths and hundredths of a foot, the thou¬ 
sandths being either estimated or omitted entirely. 

The head H on the crest of the weir is in all cases to be meas¬ 
ured several feet upstream from the crest, as indicated in Fig. 
60 c. This is necessary because of the curve taken by the surface 
of the water in approaching the weir. The distance to which 
this curve extends back from the crest of the weir depends upon 
many circumstances (Art. 70 ), but it is generally considered that 
perfectly level water will be found at 2 or 3 feet back of the crest 
for small weirs, and at 6 or 8 feet for very large weirs. It is de¬ 
sirable that the hook should be placed at least one foot from the 
sides of the feeding canal, if possible. As this is apt to render the 
position of the observer uncomfortable, some experimenters have 
placed the hook in a pail a few feet away from the canal, the water 
being led to the pail by a pipe which joins the feeding canal sev¬ 
eral feet back from the crest, and the water should enter this pipe, 
not at its end, but through a number of holes drilled at inter¬ 
vals along its circumference. Piezometers (Art. 36 ) consisting 
of a glass tube and scale are also sometimes used for large heads, 
the water being led to the tube by such a pipe. A rough method 
of measuring the head is to hold a common foot rule on a post set 
with its top on the same level as the crest and upstream from it. 

In a case where it is desired to obtain the highest degree of 
accuracy care should be taken to reproduce as nearly as possible 
the conditions which obtained under the experiments from which 
the coefficients to be used were obtained. This is particularly 
true of the manner in which the head is to be measured. Thus 
Poncelet and Lesbros, whose experimental results have been 


144 


Chap. 6 . Flow of Water over Weirs 


recomputed by Hamilton Smith, measured the head in a reservoir 
11.48 feet upstream from the weir. Francis * in some of his 
experiments measured the head with a hook gage in a wooden 
stilling box, having a hole one inch in diameter in its bottom which 
was placed at a level of about four inches below the crest of the 
weir and about 6 feet upstream from it. Fteley and Stearns f 
measured the head with a hook gage in a pail placed below the 
weir, the pail being connected to the channel above the weir at 
a point 6 feet upstream from the crest. Bazin J in his work on 
standard thin-edged weirs measured the head in pits 16.40 feet 
upstream from the weir. One pit was placed on each side of the 
channel of approach and connected with it through an opening 
4 inches in diameter, the opening being exactly flush and at right 
angles to the channel. 

A valuable discussion by Horton,§ in which he tabulates the 
results of many experiments made on weirs up to 1907, is strongly 
recommended for reference. 

In cases where the flow of water to be measured is constant it is 
best that a number of observations of the head on the measuring weir 
should be taken and their mean used in computing the quantity. 
In most practical cases, however, the flow is constantly fluctuating, 
and, in order that the total quantity may be accurately determined, 
observations at frequent intervals must be taken. It may be best 
in some cases, for convenience or where a high degree of refinement 
is required, to install an instrument such as that described in Art. 34 
for automatically and continuously recording the head. Where such 
a record has been obtained, it will not do to simply average the heads 
and use the resulting figure in the formula for the discharge. Since 
the discharges vary with the three-halves power of the head, it is 
necessary to compute them for various instants which are so selected 
that the computed discharges can be fairly averaged before multiply¬ 
ing by the total time between the beginning and end of the tests in 
order to obtain the total quantity which has passed over the weir. 
No definite rules can be laid down for this procedure, but every case 

* Lowell Hydraulic Experiments (4th edition, New York, 1883). 

f Transactions American Society of Civil Engineers, vol. 12. 

f Translated in Proceedings of Engineers Club, Philadelphia, vols. 7, 9, 10. 

§ Water Supply and Irrigation Paper No. 200, U. S. Geological Survey. 


Formulas for Discharge. Art. 61 


145 


should be studied and a plan be adopted which will give the results 
desired with the required degree of accuracy. 

Prob. 60 . The trough of a weir, several feet back from the crest, is 
6 feet wide, and the depth of water in it is 1.96 feet. What is the mean 
velocity in this trough when the flow over the weir is 4.24 cubic feet per 
second ? 

Art. 61 . Formulas for Discharge 

Referring to the demonstration of Art. 48 it is seen that a 
rectangular orifice becomes a weir when the head on its top is 
zero. Let b be the breadth of the notch, commonly called the 
length of the crest, and H the head of water on the crest. Then 
replacing hi by o and h 2 by H, the theoretic discharge per second is 

Q = I V 2 g • bIP ( 61 ): 

The head H is not the depth measured in the vertical plane of the 
crest, for since the deduction of the formula assumes nothing re¬ 
garding the fall due to the surface curve, and regards the velocity 
at any point vertically over the crest as due to the head 
upon that point below the free water surface, it seems that H 
should be measured with reference to that surface, as is actually 
done by the hook gage. The above formula then gives the 
theoretic discharge per second, provided that there be no velocity 
at the point where II is measured, which can only be the case 
when the area of the weir opening is very small compared to that 
of the cross-section of the feeding canal. This condition would 
be fulfilled for a rectangular notch at the side of a large pond. 

When there is an appreciable velocity of approach of the water 
at the point where H is measured by the hook gage, the above 
formula must be modified. Let v be sm 
the mean velocity in the feeding canal 5 -i-e: 
at this section; this velocity may be IfH 
regarded as due to a fall, //, from the 
surface of still water at some distance 
upstream from the hook, as shown in 
Fig. 61 . Now the true head on the crest of the weir is II + //, 
since this would have been the reading of the hook gage had it 
been placed where the water had no velocity. Hence the theo- 
























146 


Chap. 6 . Flow of Water over Weirs 


retie discharge per second over the weir is 

Q = l^-b(H + h) i 

in which II is read by the hook and h is to be determined from 
the mean velocity v. 

The actual discharge is always less than the theoretic dis¬ 
charge, due to the contraction of the stream and the resistances 
of the edges of the weir. To take account of these a coefficient 
is applied to the theoretic formulas in the same manner as for 
orifices; these coefficients being determined by experiment, the 
formulas may then be used for computing the actual discharge. 
It was also proposed by Hamilton Smith to modify the head /?, 
owing to the fact that the velocity of approach is not constant 
throughout the section, but greater near the surface than near 
the bottom, as in conduits and streams (Art. 125 ). Accordingly 
the following is an expression for the actual discharge: 

q = c • f V 2g • b {II + nh) 2 ( 61 h 

in which c is the coefficient of discharge whose value is always 
less, than unity, and n is a number which lies between i.o and 1.5. 
For the English system of measures a mean value of V 2g is 8.020, 
but a more precise value can be found from (6)1 for any locality. 

The above formulas are not in all respects perfectly satisfactory, 
and indeed many others have been proposed, one of these being de¬ 
rived from (50)4 by making // 0 = //, h 2 = H , and /q = o. The actual 
discharge differs, however, so much from the theoretical that the 
final dependence must be upon the coefficients deduced from experi¬ 
ment, and hence any fairly reasonable formula may be used within 
the limits for which its coefficients have been established. In spite 
of the objections which may be raised against $ill forms of formulas, 
the fact remains that the measurement of water by weirs is one of the 
most convenient methods, and for many conditions the most precise 
method. If the quantity is so small as to pass through a circular orifice 
less than one foot in diameter, then the orifice is more precise 
than the weir. For the continuous measurement of water passing 
through large pipes the Venturi meter gives the best results. With 
proper precautions the probable error in measurements of discharge 
by weirs should be less than two or three percent. 


Velocity of Approach. Art. 62 


147 


Prob. 61 . Show by using formula ( 61 ) i that an error of about one-half 
of one percent results in the computed discharge if an error of o.ooi feet 
is made in reading the head when H = 0.3 feet. 


Art. 62 . Velocity of Approach 


The head h which produces the velocity v is expressed by 
v 2 / 2 g, and in the case of a weir, the velocity of approach v is due 
to a fall from the height h ; thus the velocity-head is 

h = v 2 / 2 g = 0.01555 V 2 

and when v is known, h can be computed. One way of finding 
v is to observe the time of passage of a float through a given dis¬ 
tance ; but this is not a precise method. The usual method is 
to compute v from an approximate value of the discharge, which 
is itself first computed by regarding v, and hence /?, as zero. This 
determination is rendered possible by the fact that v is usually 
small, and hence that h is quite small as compared with H. 


Let B be the breadth of the cross-section of the feeding canal 

at the place where the readings of the hook are taken, and let 

G be its depth below the crest (Fig. 61 ). The area of that cross- 

section then is . „ , TT \ 

A = B [G + II) 


The mean velocity in this section now is 

v = q' /A 

in which the discharge q is found from the formula 

q' = c f Vig • bH- 

This value of q' is an approximation to the actual discharge; 
from it v is found, and then //, after which the discharge q can be 
computed. If thought necessary, h may be recomputed by using 
q instead of q ; but this will rarely be necessary. 

For example, a small weir with end contractions, which was 
used in the hydraulic laboratory of Lehigh University prior to 
1896, had B = 7.82 feet and G = 2.5 feet. The length of the 
weir b was adjustable according to the quantity of water deliv¬ 
ered by the stream. On April 10, 1888, the value of b was 
1.330 feet, and values of H ranged from 0.429 to 0.388 feet. 


148 


Chap. 6 . Flow of Water over Weirs 

It is required to find the velocity v and the head h, when H = 
0.429 feet. Here the coefficient c is 0.602 (Table 63 ); hence 
the approximate discharge per second is 

q' = 0.602 X f X 8.02 X 1.33 X 0.429 2 
or q' = 1.203 cubic feet per second. 

The mean velocity of approach then is 


v = --= 0.053 feet per second, 

(2.5 + 0.4)7.82 

and the head h producing this velocity is 

h = 0.01555 X 0.053 2 = 0.00004 feet, 

which is too small to be regarded, since the hook gage used 
determined the heads only to thousandths of a foot. 


The head h may be directly expressed in terms of the dis¬ 
charge by substituting for v its value q/A ; thus 

h = 0.01555 (q/A ) 2 ( 62 ) 1 

and when q is approximately known, this expression will be found 
a very convenient one for computing the value of the head cor¬ 
responding to the velocity of approach. 


The head h may be directly computed, when it is small com¬ 
pared with H, from the formula 


h 


jj ( 2 cHb V 

\ 3 (H + G)b) 



To deduce this, let the above values of A and q' be inserted in 
the equation v = q'/A , and then v be placed in h = v 2 /2g. This 
is a convenient expression for logarithmic computation. 

With a weir opening of given size under a given head II, the velocity 
of approach is less the greater the area of the section of the feeding 
canal, and it is desirable in building a weir to make this area large 
so that the velocity v may be small. For large weirs, and particularly 
for those without end contractions, v is sometimes as large as one foot 
per second, giving h — 0.0155 f ee b and these should be regarded as 
the highest values allowable if precision of measurement is required. 




Weirs with End Contractions. Art. 63 


149 


Prob. 62 . Fteley and Stearns’ large suppressed weir had the following 
dimensions: b = B = 18.996 feet, G = 6.55 feet, and the greatest measured 
head was 1.6038 feet. Taking c = 0.622, compute the velocity of approach 
and its velocity-head. 

Art. 63. Weirs with End Contractions 

Let b be the breadth of the notch or length of the weir, H 
the head above the crest measured by the hook gage, and c an 
experimental coefficient. Then, when there is no velocity of 
approach, the discharge per second is 

q = c-lWTg-bH i (63), 

But when the mean velocity of approach at the section where 
the hook is placed is v, let h be the head which would produce 
this velocity as computed by ( 62 ) 2 . Then the discharge is 

g = c-iV 7 g'b(H + i. 4 h) i ( 63) 2 

The quantity H + 1.4 h is called the effective head on the crest, 
and, as shown in the last article, the velocity-head h is usually 
small compared with the head H. 

Table 63 contains values of the coefficient of discharge c as 
deduced by Hamilton Smith, from a discussion of the experi¬ 
ments made by Lesbros, Francis, Fteley and Stearns, and others 
on standard weirs.* In these experiments q was determined by 
actual measurement in a tank of large size, and the other quan¬ 
tities being observed, the coefficient c was computed. Values of 
c for different lengths of weir and for different heads were thus 
obtained, and after plotting them mean curves were drawn from 
which immediate values were taken. The heads in the first 
column are the effective heads H + 1.4 h ; but as h is small, little 
error can result in using H as the argument with which to enter 
the table in selecting a coefficient. 

It is seen from the table that the coefficient c increases with 
the length of the weir, which is due to the fact that the end con¬ 
tractions are independent of the length. The coefficient also 


* Hamilton Smith, Hydraulics, 1884, p. 132. 


150 


Chap. 6 . Flow of Water over Weirs 


increases as the head on the crest diminishes. The table also 
shows that the greatest variation in the coefficients occurs under 
small heads, which are hence to be avoided in order to secure 
accurate measurements of discharge. 


Table 63 . Coefficients for Contracted Weirs 


ESective 
Head 
in Feet 

Length of Weir in Feet 

0.66 

I 

2 

3 

5 

10 

*9 

O.I 

0.632 

0.639 

0.646 

0.652 

0-653 

0.655 

0.656 

O.15 

.619 

.625 

•634 

.638 

.640 

.641 

.642 

0.2 

.611 

.618 

.626 

.630 

.631 

•633 

•634 

0.25 

.605 

.612 

.621 

.624 

.626 

.628 

.629 

o -3 

.601 

.608 

.616 

.619 

.621 

.624 

.625 

0.4 

•595 

.601 

.609 

.613 

.615 

.618 

.620 

o -5 

•590 

• 59 ^ 

.605 

.608 

.611 

.615 

.617 

0.6 

.587 

•593 

.601 

•605 

.608 

.613 

.615 

0.7 


•590 

•598 

•603 

.606 

.612 

.614 

0.8 



•595 

.600 

.604 

.611 

.613 

0.9 



•592 

•598 

.603 

.609 

.612 

1.0 



•590 

•595 

.601 

.608 

.611 

1.2 



•585 

• 59 i 

•597 

.605 

.610 

i -4 



• 580 

•587 

•594 

.602 

.609 

1.6 




• 582 

• 59 i 

.600 

.607 


Interpolation may be made in this table for heads and lengths 
of weirs intermediate between the values given, regarding the 
coefficient to vary uniformly between the values given. When 
coefficients are frequently required for a weir of given length, it 
will be best to make out a special table for that weir and to dia¬ 
gram the results to a large scale on cross-section paper, so that 
interpolation for different heads can be more readily made. 

As an example of the use of the formulas and Table 63 , let 
it be required to find the discharge per second over a weir 4 
feet long when the head II is 0.457 feet, there being no velocity 
of approach. From the table the coefficient of discharge is 
0.614 for II = 0.4 and 0.6095 f° r H = 0.5, which gives about 
0.612 when H = 0.457. Then the discharge per second is 

s 

q = 0.612 X § X 8.02 X 4 X 0.457“ = 4 °4 cubic feet. 
































Weirs with End Contractions. Art. 63 


151 


If the width of the feeding canal be 7 feet, and its depth below 
the crest be 1.5 feet, the velocity-head is 

h = o-oi555 ( ~t° 4 * ) = °- OOI 34 feet. 

\7 X 1.90/ 

The effective head now becomes H + 1.4 h — 0.459 feet, and the 
discharge per second over the weir is 

q = 0.612 X j X 8.02 X 4 X 0.459 2 = 4-°7 cubic feet. 

It is to be observed that the reliability of these computed dis¬ 
charges depends upon the precision of the observed quantities 
and upon the coefficient c ; this is probably liable to an error of 
one or two units in the third decimal place, which is equivalent 
to a probable error of about three-tenths of one per cent. On 
the whole, regarding the inaccuracies of observation, a probable 
error of one per cent should at least be inferred, so that the value 
q = 4.07 cubic feet per second should strictly be written q = 4.07 
± 0.04; that is, the discharge per second has 4.07 cubic feet for 
its most probable value, and it is as likely to be between the values 
4.03 and 4.11 as to be outside of those limits. 

When velocity of approach is considered, an excellent method 
of computing the discharge is to expand the parenthesis of ( 63) 2 ir 
a series and use only two terms of the expansion, thus 

( 63) 3 

in which h/ II is computed from the expression (2 cHb/$ (H + G) B)~, 
where B is the breadth of the feeding canal and G is the distance of 
the bottom of the canal below the level of the crest (Fig. 61 ). For 
example, in the case of the last paragraph h/H is found from the 
numerical data to be 0.00297, whence the quantity in the parenthesis 
is 1.00624 and the discharge is 4.04 X 1.00624 = 4.07 cubic feet per 
second. It is seen that this method requires less numerical work 
than that of the one explained above. 

In very precise* work the value of the acceleration g should be 
computed from formula (6)1 for the particular latitude and elevation 
above sea level where the weir is located. 


= c ‘ § V22 • bIP (1 + 





152 


Chap. 6 . Flow of Wafer over Weirs 


Prob. 63 . A weir in north latitude 40° 24' and 395 feet above sea level 
has a length of 2.5 feet. Compute the discharges over it, the feeding canal 
having the width 6 feet and the depth below crest 1.6 feet, when the heads 
on the crest are c.314, 0.315, and 0.316 feet. 

Art. 64. Weirs without End Contractions 

For weirs without end contractions, or suppressed weirs as 
they are often called, when there is no velocity of approach, 
the discharge per second is 

q = c-%VTg- bIP • (64), 

and when there is velocity of approach, 

q = C' \ V 2 g • b(H + i\hf (64) 2 

Here the notation is the same as in the last article, and c is to be 
taken from Table 64 , which gives the coefficients of discharge 
as deduced by Smith, in 1888. 


Table 64 . Coefficients for Suppressed Weirs 


Effective 



Length of Weir in 

Feet 



Head 








in Feet 

19 

10 

7 

5 

4 

3 

2 

O.I 

0.657 

0.658 

0.658 

O.659 




O.I 5 

•643 

.644 

.645 

•645 

0.647 

O.649 

O.652 

0.2 

•635 

•637 

■637 

.638 

.641 

.642 

•645 

O.25 

.630 

.632 

•633 

■634 

.636 

.638 

.641 

0-3 

.626 

.628 

.629 

.631 

•633 

.636 

•639 

0.4 

.621 

.623 

.625 

.628 

.630 

•633 

.636 

0.5 

.619 

.621 

.624 

.627 

.630 

•633 

•637 

0.6 

.618 

.620 

.623 

.627 

.630 

•634 

.638 

0.7 

.618 

.620 

.624 

.628 

.631 

•635 

.640 

0.8 

.618 

.621 

.625 

.629 

•633 

•637 

•643 

0.9 

.619 

.622 

.627 

.631 

•635 

•639 

•645 

1.0 

.619 

.624 

.628 

•633 

•637 

.641 

.648 

1.2 

.620 

.626 

.632 

.636 

.641 

.646 


1.4 

.622 

.629 

•634 

.640 

.644 



1.6 

.623 

.631 

•637 

.642 

.647 




It is seen that the coefficients for suppressed weirs are greater 
than for those with end contractions; this of course should be the 
case, since contractions diminish the discharge. They decrease 



















Weirs without End Contractions. Art. 64 


153 


with the length of the weir, while those for contracted weirs 
increase with the length. Their greatest variation occurs under 
low heads, where they rapidly increase as the head diminishes. 
It should be observed that these coefficients are not reliable for 
lengths of weirs under 4 feet, owing to the few experiments which 
have been made for short suppressed weirs. Hence, for small 
quantities of water, weirs with end contractions should be built in 
preference to suppressed weirs. For a weir of infinite length it 
would be immaterial whether end contractions exist or not; hence 
for such a case the coefficients lie between the values for the 19- 
foot weir in Table 63 and those for the 19-foot weir in Table 64 . 

For a numerical illustration a suppressed weir having the same 
dimensions as in the example of the last article will be used, 
namely, b = 4 feet, G = 1.5 feet, and H = 0.457 ^ eet - The co¬ 
efficient is found from Table 64 to be 0.630; then for no velocity 
of approach the discharge per second is 

3 

q = 0.630 X f X 8.02 X 4 X 0.457 2 = 4.16 cubic feet. 

Here the width B is also 4 feet; the head corresponding to the 
velocity of approach then is by ( 62 )i 

h = o.oi 555 ( = °-°°44 feet > 

\4 X 1.96/ 

and the effective head on the crest is 

H + i\h = 0.463 feet, 

from which the discharge per second is 

3 

q = 0.630 X f X 8.02 X 4 X 0.463^ = 4.24 cubic feet.. 

This shows that the velocity of approach exerts a greater in¬ 
fluence upon the discharge than in the case of a weir with end 

contractions. 

When velocity of approach exists, a good method of computation 
is to expand the parenthesis of ( 64) 2 in a series and use only two terms 
of the expansion thus, __ ,. 

q = c- iV2g •6ffMi + 2.o-J 




154 


Chap. 6 . Flow of Water over Weirs 

in which h/H can be computed from the equivalent expression 
(2 cH/$(H -fiG)). 2 For example, from the above data the value of 
h/H is 0.0095, whence the quantity in the parenthesis is 1.019 and 
q = 4.16 X 1.019 = 4.24 cubic feet per second. 

Prob. 64 . Compute the discharge per second over a weir without end 
contractions when b = 0.995 feet, H = 0.7955 feet, G = 4.6 feet. 


Art. 65 . Francis’ Formulas 


M k- 0.025 


The formulas most extensively used for computing the flow 
through weirs are those established by Francis in 1854* from the 

discussion of his numerous and 
carefully conducted experiments, 
but as they are stated without 
tabular coefficients they are to be 
regarded as giving only mean ap¬ 
proximate results. The experi¬ 
ments were made on large weirs, 
most of them 10 feet long, and 
with heads ranging from 0.4 to 1.6 
feet, so that the formulas apply 
particularly to such, rather than 
to short weirs and low heads. 
The shape and details of the crest 
of the weirs are shown in Fig. 65 
and the head was measured as described in Art. 60 . The length 
b and the head II being expressed in feet, the discharge per second, 
when there is no velocity of approach, is, for weirs without end 
contractions, or suppressed weirs, 



q = 3-33 bH ‘ 

and for weirs with two end contractions, 

q = 3-33 ( b ~ °- 2 H)H‘ 


( 65 ) 1 


( 65) 2 


Here it was considered by Francis that the effect of each end 
contraction is to diminish the effective length of the weir by 


* Lowell Hydraulic Experiments (4th edition, New York, 1883), p. 133. 















Other Weir Formulas. Art. 66 


155 


o.i II. In these formulas b and H must be taken in feet and q 
will be found in cubic feet per second. 

It is seen that the number 3.33 is c • §V2g, where c is the true 
coefficient of discharge. The 88 experiments from which this 
mean value was deduced show that the coefficient 3.33 actually 
ranged from 3.30 to 3.36, so that by the use of the mean value 
an error of one per cent in the computed discharge may occur. 
When such an error is of no importance, the formula may be safely 
used for weirs longer than 4 feet and heads greater than 0.4 feet. 


Francis’ method of correcting for velocity of approach differs 
from that of Hamilton Smith, and is the same as that explained in 
Art. 50 . The head h causing the velocity of approach is computed 
in the usual way, and then the formulas are written, for weirs without 

end contractions, „ 3 3, 

? = 3-3 + 


and for weirs with end contractions, 

q = 3.33 (b — 0.2 H)[(H + h)' 1 — h 2 ] 

It is necessary that this method of introducing the velocity of ap¬ 
proach should be strictly observed, since the mean number 3.33 was 
deduced for this form of expression. 

Prob. 65 . What modification would you introduce in (60)2, if the 
weir has one end with and the other end without contraction ? 


Art. 66. Other Weir Formulas 


Fteley and Stearns* in the discussion of their experiments 
on standard weirs proposed the formula 

(7 = 3.33 bIP + 0.007 b (66)1 

in which correction for end contraction is made as in the Francis 

formula (Art. 65 ). They also proposed the following corrections 

for velocity of approach for use in the above formula f66)i. 

„,2 ^2 

H + It = H + 1.50— H+h=H+ 2.05 — 

2 g 2g 


* Transactions American Society of Civil Engineers, vol. 12. 


156 


Chap. 6 . Flow of Water over Weirs 


the former of which is applicable to suppressed weirs and the 
latter to weirs having end contractions, v being the mean velocity 
of approach. 

Among the most recent formulas for the flow over weirs 
are those of Bazin* who experimented on sharp crests varying 
in height from 0.79 to 3.72 feet and in length from 1.64 to 6.56 
feet. From his discussion of his own results as well as those of 
Fteley and Stearns, he deduced the following formulas for weirs 
without end contractions 

Q = fi V 2g • bll 1 and Q = m^/2g-bH z (66)2 


The first of these formulas is applicable to cases where there is 
no velocity of approach, while the second, by means of the co¬ 
efficient m, corrects for any approach velocity which may exist. 
The relations between m, and H are 


m = /x 


1 -fo.s 


H 




kG + HJ j 


/X = 0.405 + 


0.00984 


where G is the height of the weir crest above the bottom of the 
channel of approach. It is thus seen that m varies with the head 
and also with the height of the weir above the bottom of the chan¬ 
nel, both of which factors influence the velocity of approach. 
On the other hand /x varies only with the head. 


Table 66. Bazin’s Coefficients m for Suppressed Weirs 


Head 


Height G of Weir Crest, 

f 

in Feet 


Feet 

0.79 

i.i 5 

1.64 

2.46 

3-72 

0.20 

O.447 

0.445 

0.444 

0.444 

0-443 

0-39 

•447 

.440 

•435 

•433 

•431 

0-59 

.458 

.446 

•438 

•432 

.427 

O.79 

.470 

•455 

•443 

•434 

.426 

0.98 

.482 

.464 

.418 

•437 

.427 

1.18 

•495 

•473 

•454 

.441 

.428 

1.38 



.460 

•444 

.429 


* Annales des ponts et chaussees, 1898; translated by Maichal and 
Trautwine in Proceedings Engineers’ Club of Philadelphia, vols. 5, 7, 
and 9. 





















Submerged Weirs. Art. 67 


157 


In the above table are given some of the values of the co¬ 
efficient m determined by Bazin’s experiments for varying heads 
and heights G of standard sharp-crested weirs. These coefficients 
are applicable only to weirs having suppressed end contractions. 
While these formulas give results agreeing well with many weir 
gagings under ordinary heads, the expression for fi cannot be re¬ 
garded as a rational one since it becomes infinite when H is zero. 

Prob. 66. What will be the value of m in the case of a weir 2.50 feet 
high when H is 1.25 feet? 

Art. 67 . Submerged Weirs 

When the water on the downstream side of the weir is allowed 
to rise higher than the level of the crest, the weir is said to be 
submerged. In such cases an entire change of condition results, 
and the preceding formulas are inapplicable. Let H be the head 
above the crest measured upstream from the weir by the hook 
gage in the usual manner, and let H' be the head above the crest 
of the water downstream from the weir measured by a second 
hook gage. If II be constant, the discharge is uninfluenced until 
the lower water rises to the level of 
the crest, provided that free access of 
air is allowed beneath the descending 
sheet of water. But as soon as it rises 
slightly above the crest so that H' has 
small values, the contraction is sup¬ 
pressed and the discharge hence increased. As II' increases, 
however, the discharge diminishes until it becomes zero when //' 
equals H. Submerged weirs cannot be relied upon to give precise 
measurements of discharge on account of the lack of experi¬ 
mental knowledge regarding them, and should hence always be 
avoided if possible. 

The following method for estimating the discharge over sub¬ 
merged weirs without end contractions is taken from the discussion 
given by Herschel* of the experiments made by Francis and by 
Fteley and Stearns. The observed head H is first multiplied 

* Transactions American Society of Civil Engineers, 1885, vol. 14, p. 194. 






















158 Chap. 6 . Flow of Water over Weirs 

by a number n, which depends upon the ratio of H' to H, and 

then the discharge is to be computed by using the modified 

Francis’ formula TT ^ / an \ 

? = 3-33 &(»#)• ( 67 )i 

The values of n deduced by Herschel * are given in Table 67 . 
They are liable to a probable error of about one unit in the second 
decimal place when II' is less than 0.2 II , and to greater errors 
in the remainder of the table, values of n less than 0.70 being 
in particular uncertain. It is seen that IF may be nearly one- 
fifth of H without affecting the discharge more than two percent. 


Table 67 . Factors for Submerged Weirs 


R' 

n 

n 

IV_ 

R 

n 

w 

R 

n 

R 

n 

0.00 

1.000 

0.18 

0.989 

00 

PO 

6 

0-935 

0.58 

0.856 

.01 

1.004 

.20 

0.985 

.40 

O.929 

.60 

0.846 

.02 

1.006 

.22 

0.980 

.42 

O.922 

.62 

0.836 

.04 

1.007 

.24 

o -975 

•44 

O.915 

.64 

0.824 

.06 

1.007 

.26 

0.970 

.46 

0.908 

.66 

0.813 

.08 

1.006 

.28 

0.964 

.48 

0.900 

• 7 o 

0.787 

.10 

1.005 

•30 

0-959 

•50 

0.892 

•75 

0.750 

.12 

1.002 

•32 

0-953 

•52 

0.884 

.80 

O.703 

• 14 

O.998 

•34 

0-947 

•54 

0.875 

.90 

0-574 

.16 

O.994 

•36 

0.941 

•56 

• 

0.866 

1.00 

0.000 


A rational formula for the discharge over submerged weirs 
may be deduced in the following manner. The theoretic dis¬ 
charge may be regarded as composed of two portions, one through 
the upper part II — IF, and the other through the lower part 
II'. The portion through the upper part is given by the usual 
weir formula, II — IF being the head, or 

Q, = l^ 7 g-b{H-H’Y 

and that through the lower part is given by the formula for a 
submerged orifice (Art. 51 ), in which b is the breadth, H r the 
height, and H — H' the effective head, or 


Q2 = bH' V 2 g (#-#') 

* Transactions American Society of Civil Engineers, 1885, vol. 14, p. 194. 






















Submerged Weirs. Art. 67 


159 


The addition of these gives the total theoretic discharge, 

Q = i^ 2 g-b(H- H'f + • bW (H - H'f 

which may be put into the more convenient form, 

Q = | • b (H + i H') (II - H'f 

The actual discharge per second may now be written, 

q = c • ! V2 } -b{H+±H') (.H - Iff 

in which c is the coefficient of discharge. 

Fteley and Stearns adoptedjthe above formula for the dis¬ 
charge, or placing M for c • § V2 g, they wrote,* 

q = Mb (H + i II') (II - H'f ‘ (67) 2 

and from their experiments deduced the following values of the 
coefficient m : 


for 

II'/II = 0.00 

0.04 

0.08 

0.12 

0.16 

0.2 

°*3 


M = 3-33 

3-35 

3-37 

3-35 

3-3 2 

3.28 

3- 21 

for 

H'/II = 0.4 

o-5 

0.6 

0.7 

0.8 

0.9 

1.0 


m = 3 -i 5 

3- 11 

3-°9 

3-09 

3.12 

3-i9 

3-33 


These are for suppressed weirs; for contracted weirs few or no 
experiments are on record. 

Thus far in this article velocity of approach has not been consid¬ 
ered. This may be taken into account in the usual way by determin¬ 
ing the velocity-head h, and thus correcting II. But it is unnecessary, 
on account of the limited use of submerged weirs, and the consequent 
lack of experimental data, to develop this branch of the subject. 
What has been given above will enable an approximate probable 
estimate to be made of the discharge in cases where the water acci¬ 
dentally rises above the crest, and further than this the use of sub¬ 
merged weirs cannot be recommended. 

Prob. 07 . Compute by the two methods the discharge over a submerged 
weir when b = 8, II = 0.46, and H =0.22 feet. 


* Transactions American Society of Civil Engineers, 1883, vol. 12, p. 103. 


160 


Chap. 6. Flow of Water over Weirs 


Art. 68. Rounded and Wide Crests 


When the inner edge of the crest of a weir is rounded as at 
A in Fig. 68, the discharge is materially increased as in the case 

of orifices (Art. 53 ), or rather 


the coefficients of discharge 
become much larger than those 
given for the standard sharp 
crests. The degree of round¬ 



ing influences so much the amount of increase that no definite 
values can be stated, and the subject is here merely mentioned 
in order to emphasize the fact that a rounded inner edge is al¬ 
ways a source of error. If the radius of the rounded edge is small, 
the sheet of escaping water is at a point below the top (a in the 
figure), which has the practical effect of increasing the measured 
head by a constant quantity. The experiments of Fteley and 
Stearns show that when the radius is less than one-half an inch, 
the discharge can be computed from the usual weir formula, seven- 
tenths of the radius being first added to the measured head H. 


Two wide-crested weirs with square inner corners are shown 
in Fig. 68, the one at B being of sufficient width so that the de¬ 
scending sheet may just touch the outer edge, causing the flow 
to be more or less disturbed, while that at C has the sheet ad¬ 
hering to the crest for some distance. In both cases the crest 
contraction occurs, although water instead of air may fill the 
space above the inner corner. For B the discharge may be 
equal to or greater than that of the standard weir having the same 
head H , depending upon whether the air has or has not free access 
beneath the sheet in the space above the crest. For C the dis¬ 
charge is always less than that of the standard weir. 

Table 68 is an abstract from the results obtained by Fteley 
and Stearns,* and gives the corrections in feet to be subtracted 
from the depths on a wide crest, like C in Fig. 68, in order to 
obtain the depths on a standard sharp-crested suppressed weir 
giving the same discharge. 

* Transactions American Society of Civil Engineers, 1883, vol. 12, p. 96. 











Rounded and Wide Crests. Art. 88 


161 


Table 68. Corrections for Wide Crests 


Head on 
Wide 

Width of Crest in Inches 

Crest 








Feet 

2 

4 

6 

8 . 

10 

12 

24 

O.05 

0.010 

0.009 

0.009 

0.009 

0.009 

0.009 

0.009 

.10 

.016 

.018 

.017 

.017 

.017 

.017 

.017 

.20 

.012 

.029 

.031 

.032 

•033 

•033 

■034 

•30 


.030 

.041 

•045 

.047 

.048 

.050 

.40 


.022 

•045 

•055 

.060 

.062 

.066 

•50 


.006 

.041 

.060 

.069 

.074 

.082 

.60 

. 


.031 

•059 

•075 

.083 

.097 

.70 



.017 

.052 

•075 

.089 

.112 

.80 



.OOO 

.040 

.071 

.091 

.125 

.90 




.027 

.062 

.089 

.137 

I.OO 




.011 

.050 

.082 

-.149 

1.20 





.021 

.061 

.168 

I.40 






.032 

0 

00 

M 


The U. S. Geological Survey* during 1903 caused to be made 
at the laboratory of Cornell University a series of experiments 
on broad-crested weirs. These experiments covered crest 
widths of from 0.479 to 1 6.302 feet and heads from 0.2 to 5.0 
feet. Without here going into detail, it was concluded from the 
results obtained that a coefficient of 2.64 may be used in the for¬ 
mula q = cbH\ for all cases of broad-crested weirs exceeding 3.0 
feet in breadth and under heads in excess of 2.0 feet. For heads of 
less than 2.0 feet the coefficients are variable and dependent on 
both the head and the width of the crest as well as on whether 
or not the nappe or water sheet remains attached to or becomes 
detached from the downstream face of the weir. For heads of 
less than 0.5 feet the sheet is very unstable and the coefficients 
fluctuate correspondingly. From 0.5 to 2.0 feet the coefficients 
are still somewhat variable and uncertain but become quite 
steady for higher heads and on crests exceeding 3.0 feet in width. 
In general when the sheet becomes detached, the coefficient be¬ 
comes equal to that for a sharp-crested weir; when the sheet is 
adherent, the coefficient may drop to 2.60. The possible range 

* Water Supply and Irrigation Paper, No. 200, U. S. Geological Survey. 






















162 Chap. 6. Flow of Water over Weirs 

in coefficients for such cases is hence seen to be from 2.60 
to 3 - 33 - 

Prob. 68. Compute the discharge for a weir like C in Fig. 68 when the 
width of crest is 1.5 feet, the head 0.85 feet, and the length of weir 10 feet. 


Art. 69 . Waste Weirs and Dams 

Waste weirs are constructed at the sides of reservoirs in order 
to allow the surplus water to escape. They are usually arranged 
so that the end contractions are suppressed. When the crest is 
narrow and the front vertical, so that the descending sheet of 
water has air upon its lower side, the discharge is approximately 
given by Francis’ weir formula (Art. 65 ), 

<7 = 3-33 

in which b is the length of the crest, and H the head measured 
some distance back from the crest. When the crest is wide and 
the approach to it is inclined, as is often the case, the discharge 
is somewhat smaller. For a crest about three feet wide and level, 
with an inclined approach back of it, Francis deduced 

q = 3.01 bH lb3 

which, for a head of one foot, gives a discharge ten percent less 
than that of the first formula. 

In constructing a waste weir the discharge q is generally known 
or assumed, and it is required to determine b and H. The latter 
being taken at 1, 2, or 3 feet, as may be judged safe and proper, 
b is found by one of these formulas. For example, let the crest 
be wide, q be 87 cubic feet per second, and H be 2.0 feet, then 

log b = log 87 — log 3.01 - 1.53 log 2 

from which log b = 1.0004, whence b = 10.o feet. When, 
however, the crest is narrow, the first formula gives b = 9.2 
feet. Evidently no great precision is needed in computing the 
length of a waste weir, since it is difficult to determine the exact 
discharge which is to pass over it, and an ample factor of safety 
should be introduced to cover unusual floods. 


Waste Weirs and Dams. Art. 69 


163 


The above formulas may be used for obtaining the approximate 
flow of a stream in which a dam with level crest has been built. 
I he water, however, is often received upon an apron of timber 
or masonry, and the inclination of this, as well as the inclination 
of the approach to the crest, materially modifies the discharge. 

The formula, _ , 

q = c • f V2g • bH 3 = ubll 1 T) 9 h 

is usually employed for dams, and it is found that the value of 
M, for English measures, may range under different circumstances 
from 2.5 to 4.2. This formula is modified below for the influence 
of velocity of approach (Art. 62 ). 

Experiments were made by Bazin in 1897* on dams from 1.6 
to 2.5 feet high with heads of water on the crests ranging from 0.2 



Fig. 69a. Fig. 69Z>. J ig. 69c. 


to 1.4 feet. For the case of Fig. 69 u the approach had an inclina¬ 
tion of 1 on 2 and the front was vertical; when the width of the 
crest was 0.33 feet, the coefficient m varied from 3.24 to 4.12 as 
the head increased from 0.27 to 1.41 feet; when the width of 
the crest was 0.66 feet, m varied from 3.10 to 3.89 for similar 
heads. For the case of Fig. 696 both approach and spron had 
slopes of 1 on 2 and the crest was 0.66 feet wide; here m increased 
from 2.83 to 3.75 as the head ranged from 0.22 to 1.42 feet. For 
Fig. 69 r, with a crest 2.62 feet wide, m ranged from 2.47 to 2.76, 
but when the upstream corner was rounded to a radius of 4 inches, 
it ranged from 2.71 to'3.12. Here it is seen that widening the 
crest decreases the discharge, as already noted in Art. 68, and that 
the apron produces a similar influence. 

Experiments on a larger scale were made by Rafter in 1898, 
for the U. S. Deep Waterways Commission at the canal of the 
Cornell hydraulic laboratory, in which the flow over dams 

* Annales cles ponts et chaussees, 1898; translated by Rafter in 

Transactions American Society of Civil Engineers, 1900, vol. 44, p. 254. 









164 Chap. 6. Flow of Water over Weirs 

was measured by a standard weir. The results of these ex¬ 
periments are given in Table 69 a, the first five being for 
darns of the form shown in Fig. 69 a, the next three for dams 
like Fig. 69 b, and the next four for dams like Fig. 69 c, those 
marked with an asterisk having the upstream corner rounded 

Table 69 a. Coefficients m for Dams 


Upstream 

Slope 

Width 
of Crest 
Feet 

Down¬ 

stream 

Slope 



Head E on Crest 

in Feet 



o -5 

1.0 

i -5 

2.0 

i 3 -o 

4.0 

5.0 

i on 2 

0-33 

Vertical 

3-35 

3.68 

3.82 

3-77 

3.68 

3-70 

3 - 7 i 

i on 2 

0.66 

Vertical 

3.22 

3-44 

3-59 

3.66 

3-68 

3 - 7 o 

3 - 7 i 

1 on 5 

0.66 

Vertical 

3 - 3 i 

3 - 3 S 

3-34 

3-35 

3-38 

3-39 

3-39 

1 on 4 

0.66 

Vertical 


3-44 

3-46 

3-48 

3-48 

3-48 

3-48 

1 on 3 

0.66 

Vertical 

3-64 

3.82 

3-83 

3-69 

3-55 

3-55 

3-55 

1 on 2 

0.00 

1 on 1 

4.21 

4.24 

4.09 

3-97 

3-83 

3-74 

3-68 

1 on 2 

0.66 

1 on 2 

3 -i 4 

3-42 

3-45 

3.61 

3.66 

3.66 

3-64 

1 on 2 

•o -33 

1 on 5 

3-30 

3-57 

3.60 

3 - 5 i 

3-47 

3-54 

3-57 

V ertical 

2.62 

Vertical 

2.60 

2.67 

2-75 

2.84 

3.01 

3.21 

3-39 

Vertical 

2.62 * 

Vertical 

2.96 

3.01 

3-03 

3.08 

3-25 

3-3 8 

3-47 

Vertical 

6.56 

Vertical 

2.50 

2.60 

2-54 

2.48 

2-51 

2.61 

2.70 

Vertical 

6.56* 

Vertical 

2.71 

2.83 

2.84 

2.84 

2.86 

2.90 

2-94 

1 on 1 

Round 

V ertical 

2-95 

3 -i 7 

3 - 3 i 

3-45 

3-56 

3.61 

3-65 


to a radius of 4 inches. The last line of the table refers to a sec¬ 
tion whose top was 5 feet wide and rounded to a radius of 3.37 
feet, the rounding beginning on the upstream side 1.00 foot 
below the crest. The height of these dams varied from 4.56 
to 4.91 feet, and the length of the crest was in all cases 6.58 feet.* 



Fig. G 9 d. Fig. 69 c. Fig. 69 /. 


Rafter also made experiments on some other forms of dams. 
The one shown in Fig. 69 </ had a vertical front 4.57 feet deep, 
and the two back slopes were i on 6 and i on f, the width of the 
former being 4.5 feet; the values of m for this case ranged from 


* Transactions American Society of Civil Engineers, 1900, vol. 44, p. 266. 



































Waste Weirs and Dams. Art. 69 165 

3.33 to 3.46 for heads ranging from 1.0 to 6.0 feet. The one 
shown in Fig. 69 ^ had a total width of about 23 feet and a height 
°f 4-53 f eet > the slopes of the approach and apron being 1 on 6, 
and that just below the crest about 1 on J, the vertical depth 
of this being 0.75 feet; for this the mean values of m ranged from 
3.07 to 3.27 for heads ranging from 1.0 to 6.0 feet, the smaller co¬ 
efficients being due to the contact of the water with the apron. 

For ogee dams similar in cross- 
section to Fig. 69 /, experiments 
were made in 1903 * by the U. S. 

Geological Survey. The widths 
a of the various crests ranged 
from 3.0 to 6.0 feet, the radii r 
from 1.0 to 3.0 feet, and the rises 
c from 0.75 feet to 2.88 feet. 

From a discussion of these results 
it was concluded that the coeffi¬ 
cient m has a value of (3.78 — 0.16 s ) H 26 , where 5 is the ratio 
of a to c in Fig. 69 g. For example, when s = 3.0/1.5 and H = 
4.0 feet, then m = 3.70. 

In the table on the next page are shown the principal re¬ 
sults of the above experiments on models of ogee dams : 

The height of the crests above the bottom of the channel of 
approach of all the models was 11.25 feet an d the heads were 
measured at two points, one 10.3 feet and the other 16.059 
feet upstream from the weir crest. It was found that in general 
the reading of the gage nearest the weir was not affected by 
the surface curve for heads of less than three feet on the crest. 
The water which was used in these experiments was measured 
over a sharp-crested standard weir 6.65 feet high and having 
a crest 15.93 feet 1 R length. 

By the use of these coefficients the discharge of a stream over 
a dam may be computed with a good degree of precision. For- 



* Water Supply and Irrigation Paper No. 200, p. 131. 








166 


Chap. 6. Flow of Water over Weirs 


Table 69 &. Coefficients m for Ogee Dams 



1 

2 

3 

4 

s 

6 

7 

8 

a , feet 
f, feet 
r, feet 

3.00* 

0-75 

3.00 

3.oof 

0-75 

3.00 

3 -°°* 

1.50 

3.00 

3.00 f 
1.50 

3.00 

3.00* 

2.88 

3.00 

4 - 50 * 

1.00 

2.00 

14 - 83 * 

1.00 

2.00 

6.00* 

1.00 

1.00 

He d in 
Feet 

^ Value of Coefficient m 

O.50 

I.OO 

2.00 

3.00 

4-00 

5.00 

3-31 

3-44 

3-4 2 
3-46 
3-52 

3-29 

3-36 

3-43 

3-53 

3-72 

3.21 

3-48 

3-67 

3-7 2 

3-74 

3 -27 

3-37 

3-5i 

3-57 

3-67 

3.82 

3-i5 

3-45 

3-75 

3-87 

3.88 

3 -i 8 

3-30 

3-42 

3-49 

3-53 

3-23 

3-34 

3-52 

364 

3-70 

3.28 

3-49 

3-42 

3 - 3 i 

3-30 


* Length of crest 15.969 feet, contractions suppressed, 
f Length of crest 7.938 feet, with one end contraction. 

X This model had upstream corner rounded to radius of 4 inches. 


mula ( 62 )! may be used to find the head corresponding to the 
velocity of approach, and then 

q = Mb (II + //)“ ( 69) 2 

gives the discharge in cubic feet per second. For example, when 
M = 3 . 45 , b = 1.50 feet, II = 1.25 feet, h = 0.02 feet, then 
q — iiio cubic feet per second. A fair estimate of the probable 
error of a coefficient M is from 3 to 4 percent. 


The following formula has been found to give good results in 

automatically applying a correction for the velocity of approach for 

heads above o. s feet. , rr i 

= M b I P 

1 H /3 (G + H) 


where G is the height of the weir crest above the bottom of the ap¬ 
proach channel. It will be noted that in form the term II/$ (G+H) 
is similar to »the correction for velocity of approach used by 
Bazin (Art. 66). 


Trob. 69 . Find the length of a waste weir which will be ample to dis¬ 
charge a rainfall of one inch per hour on a drainage area of 3.65 square miles, 


































The Surface Curve. Art. 70 


167 


the head on the crest of the weir being 2.12 feet. Also when the head is 
4.24 feet. 


Art. 70 . The Surface Curve 


The surface of the water above a weir or dam assumes a curve 
whose equation is a complex one, but some of the laws that govern 
the drop in the plane of the crest may be deduced. 

Let II be the head on the level of the crest meas¬ 
ured in perfectly level water at some distance 
back of the weir, and let d be the depression or 
drop of the curve below this level in the plane of 
the weir (Fig. 70 ). Then the discharge per sec¬ 
ond q can be expressed in terms of II and d by 
formula ( 50 ) 4 , placing II for // 2 and d for h h and 
making h Q = o. This formula becomes, after replacing f V2g by m, 

and Q h y q = M. ‘ b (fl* - <fi) 

This expression, it may be remarked, is the true weir formula, and only 

the practical difficulties of measuring II and d prevent its use. This 

may be written 3 3 

d 2 = H 2 — q/ub 



Fig. 70. 


from which the drop d in the plane of crest of the weir can be found. 
Let B be the breadth of the feeding canal, G its depth below the crest, 
and v the mean velocity of approach; then also 

q = B (G + H)v 

3 

and inserting this in the expression for d 2 it becomes 

( fi = Hi- -- (C, + H)v ( 70 ) 

m b 


which is an expression for the drop of the curve in terms of the dimen¬ 
sions of the weir, the total head, and the velocity of approach. 

The approximate value of the coefficient m is about 3.3 for English 
measures, but precise values of d cannot be computed unless m and 
II are known with accuracy. The formula, however, serves to ex¬ 
emplify the laws which govern the drop of the curve in the plane of the 
weir. It shows that the drop increases with the head on the crest 
and with the length of a contracted weir, that it decreases with the 
breadth and depth of the feeding canal, and that it decreases with the 
velocity of approach. It also shows for suppressed weirs, where B = b, 


















168 


Chap. 6 . Flow of Water over Weirs 


that the drop is independent of the length of the weir. All of these 
laws except the last have been previously deduced by the discussion 
of experiments. 

The path of the stream after leaving the weir is closely that of 
a parabola. In the plane of the crest the mean velocity is 

V = q/b(H - d) 


and the direction of this may be taken as approximately horizontal. 
The range of a stream on a horizontal plane at the distance y below 
the middle of the weir notch is then readily found. For, if x be this 
range which is reached in the time /, then x = Vt, and also y = i gf; 
whence, by the elimination of /, there results gx 2 = 2 V 2 y, and accord¬ 
ingly the horizontal range at the depth y is 


x = M 


3 3 /- 

2_y 

H-d\g 


in which d is given by ( 70 ). For example, take a case where H = 3 
feet, G = 23 feet, and v = 0.5 feet per second. From ( 70 ) the value 
of d is found to be 1.17 feet. Now, when y = 50 feet, the last formula 
gives x = 12.5 feet, which is the horizontal distance of the middle of 
the stream from the vertical plane through the crest. 


Prob. 70 . In the above example what velocity of approach is necessary 
in order that there may be no drop in the plane of the crest ? What is the 
range for this case? 


Art. 71 . Triangular Weirs 


Triangular weirs are sometimes used for the measurement 
of water, the arrangement being shown in Fig. 71 . Let b be 

the width of the orifice at the 
water level, and H the head of 
water on the vertex. Let an 
elementary strip of the depth By 
be drawn at a distance y below 
the water level. From similar 
triangles the length of this strip is (H — y)b/H and the elemen¬ 
tary discharge through it then is 

ZQ = jj(H-y) b ^2gy = ~b2g ( Hy i - y^Sy 
















Triangular Weirs. Art. 71 


169 


The integration of this between the limits H and o gives the the¬ 
oretic discharge through the triangular weir, namely, 

Q = -&b^/Tg-H l ( 71 ), 

If the sides of the triangle are equally inclined to the vertical, 
as should be the case in practice, and if this angle be a, the sur¬ 
face width b may be expressed in terms of a and H, so that the last 

formula becomes ,— a 

£> = &tan«- V2g-H s ( 71 )* 

The discharge is thus equal to a constant multiplied by the 2J 
power of the measured depth. 

• 

Triangular weirs are used but little, as in general they are 
only convenient when the quantity of water to be measured is 
small. Such a weir must have sharp inner corners, so that the 
stream may be fully contracted, and the sides should have equal 
slopes. The angle at the lower vertex should be a right angle, 
as this is the only case for which coefficients are known with pre¬ 
cision. The depth of water above this lower vertex is to be meas¬ 
ured by a hook gage in the usual manner at a point several 
feet upstream from the notch. Making the angle at the vertex 
a right angle, and applying a coefficient, the actual discharge 
per second is given by the expression 

q = c-j% V2g jp 

in which H is the head of water above the vertex. Experiments 
made by Thomson * indicate that the coefficient c varies less 
with the head than for ordinary weirs; this, in fact, was antici¬ 
pated, since the sections of the stream are similar in a triangular 
notch for all values of H, and hence the influence of the contrac¬ 
tions in diminishing the discharge should be approximately 
the same. As the result of his experiments the mean value of 
c for heads between 0.2 and 0.8 feet may be taken as 0.592, and 
hence the mean discharge in cubic feet per second through a 
right-angled triangular weir may be written 

2=2.53# 2 ( 71)3 

* British Association Report, 1858, p. 133. 


170 Chap. 6. Flow of Water over Weirs 

in which, as usual, II must be expressed in feet. About 4 feet 
is probably the greatest practicable value for II, and this gives 
a discharge of only 81 cubic feet per second. When velocity 
of approach exists, H in this formula should be replaced by II + 
1.4//, as for rectangular weirs with end contractions. 

Prob. 71 . A triangular orifice in the side of a vessel has a horizontal 
base b and an altitude d , the head of water on the base being h and that on 
the vertex being li 4 - d. Show that the theoretic discharge through the 

orifice is -h\Z 2 g(b/d) • [4(A -(4 h + 10 d)h 2 ]. 

Art. 72 . Trapezoidal Weirs 

Trapezoidal weirs are sometimes used instead of rectangular 
ones, as the coefficients vary less in value. The theoretic 

discharge through a trapezoidal 
weir which has the length b on 
the crest, the head II, and the 
length b + 2s on the water sur¬ 
face, as seen in Fig. 72 , is the sum 
of the discharges through a rect¬ 
angle of area bll and a triangle of area zH. Taking the former 
from (61 )i and the latter from (71 ) 2 , and replacing tan« by z/H 

Q = AVii(s6 + AZ)IP 

is the theoretic discharge. Here z/H, which is the slope of the 
ends, may be any convenient number, and it is usually taken as 
I, as first recommended by Cippoletti.* 

The reasoning from which this conclusion was derived is 
based upon Francis’ rule that the two end contractions in a 
standard rectangular weir diminished the discharge by a mean 

5 

amount 3.33 X 0.2 H a (Art. 65 ), or in general by the amount 
c * § V2 g X 0.2 II-i. If the sides are sloped, however, the discharge 
through the two end triangles is c • V2 g X zIlL If, now, the 
slope is just sufficient so that the extra discharge balances the 
effect of the end contractions, these two quantities are equal. 
Equating them, and supposing that c has the same value in each, 



* Cippoletti, Canal Villoresi, 1887; see Engineering Record, Aug. 13, 1892. 


































Trapezoidal Weirs. Art. 72 


171 


there results z = \II. Hence for such a trapezoidal weir the dis¬ 
charge should be the same as that from a suppressed rectangular 
weir of length b, or, according to Francis, q = 3.33 bHl. Cip- 
poletti, however, concluded from his experiments that the coeffi¬ 
cient should be increased about one percent, and he recommended 

q = 3.367 hW ( 72 ) 

as the formula for discharge over such a trapezoidal weir when 
no velocity of approach exists. 

Experiments by Flinn and Dyer* indicate that the coefficient 
3.367 is probably a little too large. In 32 tests with trapezoidal 
weirs of from 3 to 9 feet length on the crest and under heads rang¬ 
ing from 0.2 to 1.4 feet, they found 28 to give discharges less than 
the formula, the percentage of error being over 3 percent in 
eight cases. The four cases in which the discharge was greater 
than that given by the formula show a mean excess of about 
3.5 percent. The mean deficiency in all the 32 cases was 
nearly 2 percent. These experiments are not very precise, 
since the actual discharge was computed by measurements on 
a rectangular weir, so that the results are necessarily affected 
by the errors of two sets of measurements. Cippoletti ’s for¬ 
mula, given above, may hence be allowed to stand as a fair one 
for general use with trapezoidal weirs in which the slope of the 
ends is J. It can, of course, be written in the form 

q = c • | VTg ■ bH l 

where the coefficient c has the mean value 0.629. 

When velocity of approach exists, H in this formula is to be re¬ 
placed by II + 1.4 h, where h is the head due to that velocity. In 
order to do good work, however, h should not exceed 0.004 f eet - 
Other precautions to be observed are that the cross-section of the 
canal should be at least seven times that of the water in the plane of 
the crest, and that the error in the measured head should not be greater 
than one-third of one percent. On the whole, however, the coefficients 
for the standard rectangular weir with end contractions are so definitely 
established, and those for trapezoidal weirs so imperfectly known, 

* Transactions American Society of Civil Engineers, 1894, vol. 32, pp. 9~ 33 - 


172 


Chap. 6 . Flow of Water over Weirs 


that the use of the latter cannot be recommended in any case where 
the greatest degree of precision is required. 


The above formula for the theoretic discharge may be applied to 

the Cippoletti trapezoidal weir by putting z = \ H, and introducing 

a coefficient: thus, 9 ,— . _s , _ T/f . 

’ ’ q = c§V2g* W 7 2 (i + 0.2 H/b) 


is a formula for the actual discharge, in which the values of c are prob¬ 
ably not far from those given in Table 63 for rectangular contracted 
weirs. Here the term 0.2 H/b shows the effect of the two end triangles 
in increasing the discharge. 

Prob. 72 . For a head of 0.7862 feet on a Cippoletti weir of 4 feet length 
the actual discharge in 420 seconds was 3912.3 cubic feet. Compute the 
discharge by the above formula, and find the percentage of error. 


Art. 73 . Oblique Weirs 

In certain cases weirs or dams are built obliquely across streams 
and in others there may be either a curve or one or more angles 
in the line of the crest. When the volume of the flow in the stream 
is small, so that the water may at all points approach the crest 
in a direction sensibly at right angles to it, the discharge will be 
proportional to the crest length and may be computed by the 
formulas already given. When, however, the flow of the stream 
becomes so great that the water approaches the crest in an oblique 
direction, the discharge tends to approximate that over a weir 
placed at right angles to the axis of the stream. This, however, 
is not strictly true in case the obliquity be material. In such a 
case the discharge for the same head is increased above that over 
a weir built normal to the axis of the stream. This condition 
is sometimes taken advantage of where it is desired to -keep 
down the effect of backwater during times of flood, but such an 
arrangement causes a loss of available head during times of me¬ 
dium and low water. The problem of the regulation of river 
heights is, under certain conditions, an important one and is 
well exemplified by the conditions at the Chaudiere Dam, Ottawa.* 

Achiel f experimented on weirs inclined to the axis of the chan- 

* Engineering News, June 30, 1910. 

t Zeitschrift Verein Deutschen Ingenieure; see abstract in Engineering 

Record, July 3, 1909. 


Computations in the Metric System. Art. 74 173 

nel at angles varying from 15 to 90°. These weirs were placed 
in channels 1.64 and 3.28 feet in width, the end contractions were 
suppressed, and the nappe was thoroughly aerated; their height 
was 0.82 feet and the heads ranged from 0.04 to 0.60 
feet. From these experiments the formula F c = 1 — H/Gr 
was deduced. Here H is the measured head on the weir, G the 
height of the weir crest above the channel of approach, and r a 
number taken from the table below. F c then is a correction 
factor by which the values of the coefficient for a vertical thin- 
edged weir are to be multiplied in order to obtain the coefficients 
for each unit of length of the oblique weir. This formula does 


Angle of weir =I 5 ° 

3 °° 

45 ° 

On 

O 

O 

75 ° 

90 

r for broad channels = 1.4 

2.8 

5 -o 

9.1 

26.3 

00 

r for narrow channels = 1.2 

2.1 

3-6 

7-7 

26.3 

00 


not hold when the ratio H/G is greater than 0.62, and this ratio 
should be smaller as the obliquity of the weir increases. In 
general it can be said that outside the range of the few experiments 
which have been made but little is known on this subject. 

Prob. 73 . What is the coefficient for an oblique sharp-edged weir 
with contractions suppressed, 19 feet long and two feet in height when the 
head is 0.6 feet and the obliquity of the weir 45 degrees ? 

Art. 74 . Computations in the Metric System 

The formulas for discharge in Arts. 61-64 are rational and may be 
used in all systems, the coefficients c being abstract numbers. In the 
metric system b and H are often expressed in centimeters, but they 
should be reduced to meters for use in the formulas, and then q will 
be in cubic meters per second. The rhean value of V2 g is 4.427 and 
that of 1/2 g is 0.05102. 

(Art. 62 ) The head h in meters corresponding to the mean veloc¬ 
ity of approach is to be computed from the formula 

h — 0.05102 (q/A ) 2 ( 74 )i 

in which A is in square meters. For example, take a weir where B 
= 200, G = 90, b = 45.1, H = 26.28 centimeters, and c = 0.620. 


174 


Chap. 6 . Flow of Water over Weirs 


Then by ( 63 )! the discharge q' is 0.1112 cubic meters per second, 
and from ( 74 )! the head h is 0.0002 meters. 

(Art. 63 ) Table 74 a gives values of the coefficient c for weirs 
with end contractions, with arguments in the metric system. Thus, 
if H = 5.45 centimeters and 6 = 0.45 meters, there is found, by in¬ 
terpolation, c = 0.626, which is liable to a probable error of about 
two units in the third decimal place. 

Table 74 a. Coefficients c for Contracted Weirs 


Arguments in Metric Measures 


Effective 
Head in 

Length of Weir in Meters 

Centi- 








meters 

0.2 

03 

0.6 

0.9 

1.5 

30 

5-8 

3 - 

O.633 

0.640 

0.647 

0.653 

0.654 

0.656 

0.657 

5 - 

.618 

.624 

.634 

.638 

.640 

.641 

.642 

7 - 

.606 

.613 

.622 

.625 

.627 

.629 

.630 

9 - 

.601 

.608 

.616 

.619 

.621 

.624 

.625 

-12. 

•596 

.602 

.609 

.613 

.615 

.618 

.620 

i 5 - 

•591 

•597 

.605 

.608 

.611 

.615 

.617 

18. 

.588 

•593 

.601 

.605 

.608 

.613 

•615 

22. 


•589 

•597 

.603 

.606 

.612 

.614 

26. 



•594 

•599 

.604 

.610 

.613 

3 °- 



•590 

•595 

.601 

.608 

.611 

35 - 



.586 

•592 

•597 

•605 

.610 

45 - 




•585 

•593 

.601 

.608 


(Art. 64 ) Coefficients c for weirs without end contractions, with 
metric arguments, are given in Table 746 , which has been prepared 
by the help of Table 64 . 

(Art. 65 ) When b and H are in meters and q in cubic meters per 
second, Francis’ formula for suppressed weirs takes the form 

q= 1.84 bIP (74)2 

and for weirs with end contractions, 

9=1.84(6 — 0.2#)//- (74)3 

the number 1.84 being a mean value of c • § 

(Art. 67 ) Table 67 applies to any system of measures, and the 
formula q = 1.84 b{nH ) 2 then gives the discharge in cubic meters per 




























Computations in the Metric System. Art. 74 


1 i o 


Table 74 b . Coefficients c for Suppressed Weirs 


Arguments in Metric Measures 


Effective 
Head in 

Length oi Weir in Meters 

Centi- 








meters 

5-8 

3 -o 

2.0 

15 

1.2 

0.9 

0.6 

3 - 

0.658 

0.659 

0.659 

0.660 




5 - 

.642 

•643 

.644 

•645 

0.647 

0.649 

0.652 

7 - 

.632 

•633 

•634 

•635 

•637 

.640 

• 643 

9 - 

.626 

.628 

.629 

.631 

•633 

.636 

•639 

12. 

.621 

.623 

.625 

.628 

.630 

•633 

.636 

i 5 - 

.619 

.621 

.624 

.627 

.630 

•633 

• 637 

18. 

.618 

.620 

.623 

.627 

.630 

•634 

.638 

22. 

.618 

.620 

.624 

.628 

.632 

.636 

.640 

26. 

.619 

.622 

.627 

.631 

•635 

•639 

• 645 

30. 

.619 

.624 

.628 

•633 

•637 

.641 


35 - 

.620 

.626 

.631 

•635 

.640 

•645 


45 - 

.622 

.630 

•635 

.641 

•645 




second, if b and H be in meters. The metric values of m for use in 
( 67) 2 are found by multiplying those in the text by 0.5522. 

(Art. 69 ) The formulas of the first paragraph are transformed 
into metric measures by replacing 3.33 by 1.84 and 3.01 by 1.72. For 
formula ( 69 )! the value of m for dams may range from about 1.4 to 
2.3. Table 74 c gives metric values of m as deduced from the experi¬ 
ments made by Bazin in 1897, and by Rafter in 1898. The explana¬ 
tion of this table is in all respects like that of Table 69 a. All values 
of m given in Art. 69 may be reduced to metric measures by multi¬ 
plying by 0.5522, this being the ratio of the value of V 2g expressed in 
meters to that expressed in feet. 

(Art. 71 ) The metric formula for discharge over the triangular 
weir is q = 1.40 Z7A 

(Art. 72 ) The metric formula for Cippoletti’s trapezoidal weir 
takes the form q — 1.86 bHK 

Prob. 74 a. Compute the head that produces a velocity of approach 
of 50.5 centimeters per second. 

Prob. 74 b. What are the discharges, in liters per minute, over a sup¬ 
pressed weir 2.35 meters long when the heads on the crest are 12.3, 12.4, 
and 12.5 centimeters? 



























176 


Chap. 6. Flow of Water over Weirs 


Table 74 c. Coefficients m for Dams 


Metric Measures 


Up¬ 

stream 

Slope 

Width 
of Crest 
Meters 

Down¬ 

stream 

Slope 

Head H on Crest in Meters 

0.15 

0.30 

0.60 

0.91 

x.22 

1-52 

1 on 2 

O.IO 

Vertical 

1.85 

2.03 

2.08 

2.03 

2.04 

2.05 

i on 2 

0.20 

Vertical 

I.78 

1.90 

2.02 

2.03 

2.04 

2.05 

1 on 5 

0.20 

Vertical 

I.83 

1.84 

1.85 

1.86 

1.87 

1.87 

1 on 4 

0.20 

Vertical 


1.90 

1.92 

1.92 

1.92 

1.92 

1 on 3 

0.20 

Vertical 

2.01 

2.11 

2.04 

1.96 

1.96 

1.96 

1 on 2 

0.00 

1 on 1 

2-33 

2-34 

2.19 

2.11 

2.06 

2.03 

1 on 2 

O.IO 

1 on 2 

i -73 

1.90 

1.99 

2.02 

2.02 

2.01 

1 on 2 

0.20 

1 on 5 

1.82 

1.97 

1.94 

i -93 

i -95 

1.97 

Vertical 

O.80 

Vertical 

i -43 

1.47 

1-57 

1.66 

1.77 

I.87 

Vertical 

0.80* 

Vertical 

1.63 

1.66 

1.70 

1.79 

1.87 

I.92 

Vertical 

2.00 

Vertical 

1.38 

i -43 

i -37 

i -39 

1.44 

I.49 

Vertical 

2.00 * 

Vertical 

1.50 

1.56 

i -57 

1.58 

1.60 

I.63 

1 on 1 

Round 

Vertical 

1.63 

i -75 

1.91 

1.96 

1.99 

2.01 


* For explanation see Art. 69 . 


Prob. 74 c. Compute the discharge over a submerged weir when b = 
2.35, H = 0.123, and H' = 0.027 meters. 

Prob. 74 d. Compute the discharge over a dam, like Fig. 686, when the 
side slopes are 1 on 2, the length of the crest 4.25 meters, and the head on 
the crest 1.07 meters. 






















Loss of Energy or Head. Art. 75 


177 


CHAPTER 7 


FLOW OF WATER THROUGH TUBES 


Art. 75 . Loss of Energy or Head 


A tube is a short pipe which may be attached to an orifice 
or be used for connecting two vessels. The most common form 
is a cylinder of uniform cross-section, but conical forms are also 
used, and in some cases a tube is made of cylinders with different 
diameters. The laws of flow through tubes are important as 
a starting-point for the theory of flow through pipes, for the dis¬ 
charge from nozzles, and for the discussion of many practical 
hydraulic problems. The theorem of Art. 31 , that pressure- 
head plus velocity-head is a constant for a given section of a 
tube, is only true when there are no losses due to friction and im¬ 
pact. As a matter of fact such losses always exist and must be 
regarded in practical computations. 

Energy in a tube filled with moving water exists in two forms, 
in potential energy of pressure and in kinetic energy of motion. 
Thus in the horizontal tube of 
Fig. 7 5 a let two piezometers 
(Art. 37 ) be inserted at the sec¬ 
tions ai and a 2 where the velocities 
are Vi and v 2 and it is found that 
the water rises to the heights h 
and hi above the middle of the 
tube. Let W be the weight of 
water that passes each section per 
second. Then in the first section the pressure energy per sec¬ 
ond is Wh\ and the kinetic energy per second is W • V\ 2 / 2 g, so 
that the total energy of the water passing that section in one 
second is whi + w . Vi y 2g 



Fig. 75 a. 
















































178 


Chap. 7 . Flow of Water through Tubes 


In the same manner the total energy of the water passing the 
second section in one second is 


Wh + W • V 2 2 / 2 g 

but this is less than the former because some energy has been 
expended in friction and impact. Let Wh! be the amount of 
energy thus lost; then equating this to the difference of the ener¬ 
gies in the two sections, the W cancels out and 

h' = hi-fh + —-‘^- ( 75 ), 

2 g 2 g 

The quantity h ' is called the lost head, and the equation shows 
that it equals the difference of the pressure-heads plus the differ¬ 
ence of the velocity-heads. 


In hydraulics the terms “ energy" and "head" are often used 
as equivalent, although really energy is proportional to head. In 
the general case, the lost head is not a loss of pressure-head only, 
but a loss of both pressure-head and velocity-head. When, 
however, the two sections are of equal area, the velocities 
and v 2 are equal, since the same quantity of water passes each 
section in one second; then the lost head h f is hi — h 2 or the loss 
occurs in pressure-head only. Here the loss is mainly due to 
the roughness of the interior surface of the tube or pipe. It 
should be noted that it is only necessary to measure the difference 
hi — li 2 and this can be done bv the methods of Art. 37 . 


Formula ( 75 )i is applicable to all horizontal tubes and pipes, 
and with a slight modification it is also applicable to inclined 

ones, as will be 
shown in Art. 85 . 
It also applies to a 
flow from a standard 
orifice, or to the flow 
from an orifice to 
which a tube is at¬ 
tached. Thus for the large vessel of Fig. 75 h let the sections 
be taken through the vessel and through the stream as it leaves 
the tube. Then hi = h, and since there is no pressure outside 




h 




Fig. 7 5 b . 


Fig. 7 o<. 































Loss Due to Expansion of Section. Art. 76 


179 


the tube, h 2 = o; also Vy = o and v 2 = v; then ti = h — v 2 /2g. 
For the case in Fig. 75 c, where the stream approaches with the 
velocity Vy , the formula becomes ti = hy + (vy 2 — v 2 )/2g. In 
both cases, if ti is made zero, these equations reduce to those 
established in the chapter on theoretical hydraulics, where losses 
of energy were not considered; thus for the second case the theo¬ 
retic effective head h is equal to hy + !’l 2 /2 g . 

In order to use ( 75 )i for numerical computations three quan¬ 
tities must be known, the difference h \— h 2 , and the velocities 
Vi and v 2 . As a direct measurement of the velocities is usually 
impracticable, these are generally computed from the measured 
discharge q and the areas ay and a 2 of the cross-settions; thus 
Vy = q/ay and v 2 = q/a 2 . For example, let the cross-section be 
circular, having diameters of 18 and 6 inches, and let the discharge 
be 4.7 cubic feet per second; the areas are 1.767 and 0.196 square 
feet, and the velocities are 2.66 and 23.94 feet per second. If 
the difference of the pressure-heads is 8.85 feet, the lost head is 


ti = 8.85 + 0.01555(2.66 2 - 23.94“) = 0.05 feet 

The general formula ( 75 ) 1 may be expressed in terms of the areas 
of the sections and one of the velocities. Since ayVy = a 2 v 2 it may 
be written / 

ti = liy - h +( 1 


ay 2 \Vy 2 


or 


ti = hy — h 2 -h 

V 


a 2 ) 
2 x) 


2g 

V 2 


ay 


J 2 g 


( 75 ), 

( 75 ) 3 


which are often convenient forms for numerical computations. 


Prob. 75 . In Fig. 75 a let the areas ay and a 2 be 1.0 and 0.5 square feet, 
hy - h 2 = 0.697 feet, and Vy = 3.5 feet per second. What is the value of 
the lost head ? 


Art. 76 . Loss Due to Expansion of Section 

When a tube or pipe is filled with flowing water a loss of head 
is found to occur when the section is enlarged, so that the velocity 
is diminished. This case is shown in Fig. 76 a, where Vy and v 2 
are the velocities in the smaller and larger sections and hy and h 2 
the corresponding pressure-heads. The interior surface may be 



180 


; 

Chap. 7 . Flow of Water through Tubes 


very smooth, so that friction has but little influence, and yet 
there will usually be more or less loss due to the fact that the veloc¬ 
ity is changed to the smaller value v 2 . Formula (/ 5 )i is here 
directly applicable and gives the loss of head. It is seen that 
hi — h 2 must be negative for this case and that its numerical 
value will be less than that of the difference of the velocity-heads. 
The general formula ( 75 )i gives the loss of head due not only to 
expansion of section, but to all resistances between any two sec¬ 
tions of a horizontal tube or pipe. 

When there is a sudden enlargement of section, as in Fig. 
766 , energy is lost in impact. In the section AB the pressure- 



head is h\ and the velocity-head is v 2 / 2 g, while in the section CD 
the pressure-head has the larger value h 2 and the velocity-head 
has the smaller value v 2 2 /2g. At the section MN, near the place 
of sudden expansion, the pressure-head is also hi, since the velocity 
Vi is maintained for a short distance after leaving the small 
section; its direction, however, being changed so as to form 
whirls and foam. In this region the impact occurs, the velocity 
Vi being finally decreased to v 2 . Let a 2 be the area of the sections 
MN and CD, and w the weight of a cubic unit of water. Then 
by ( 15 ) the hydrostatic pressure normal to the section CD is 
wa 2 h 2 , and that normal to the section MN is wa 2 h\. The dif¬ 
ference of these pressures is the force which causes the veloc¬ 
ity Vi to decrease to v 2 , and by Art. 27 , this force is equal to 
W(vi—v 2 )/g, where W is the weight of water passing the section 
CD in one second. Hence 












































































Loss Due to Expansion of Section. Art. 76 


181 


waohi — waohi = W —-— 

g 

and, since W equals wcw 2 , this equation becomes 

h - h = ( 76 ), 

g 

Inserting this value of h ~ h in ( 75 )i, it reduces to 

• h - __ fa - V2) 2 
2 g 

which is the loss of head due to sudden expansion of section, or 
rather due to the sudden diminution of velocity that is caused 
by such expansion. 

When the expansion of section is made gradually and with 
smooth curves, the velocity vi will decrease without whirl and 
foam, so that no loss in impact occurs. In this case the kinetic 
energy w • v 2 /2g is changed into pressure energy, as the velocity 
Vi decreases to v 2 . There is, however, no distinct line of demar¬ 
cation between sudden and gradual expansion, so that in many 
practical cases it is necessary to make measurements of the dis¬ 
charge and of the head h 2 — h\ in order to compute the lost head 
h' from ( 75 ) 1, which is a formula applicable to all cases. 

Sudden enlargement of section should always be avoided in 
tubes and pipes owing to the loss of head that it causes, which 
may often be very great. For example, let there be no pressure- 
head in the section di and let V\ be due to a head h so that Vi = 
V2 gh ; let the area d 2 be four times that of d\ so that v 2 is one- 
fourth of V\. The loss of head due to sudden expansion then is 



fa - Pg ) 2 _ JL h 
16 n 


2g 


so that more than one-half of the energy of the water in a\ is 
lost in impact, having been changed into heat. In the section 
d 2 the effective head is T 7 g h , of which y 1 ^ h is velocity-head and 
y 6 e h is pressure-head. 

Formula ( 76 )i may be expressed in terms of the areas of the 







182 


Chap. 7 . Flow of Water through Tubes 


sections and one of the velocities, since d\V\ = doVz. The value 
of h' takes the two forms 


~ , 9 9 


V ( 1 * 2 ' 2 g \di 


, 2 Z »‘> 2 


2g 


( 76 ), 


and these show that no loss of head occurs when ai = a* 2 . 


Prob. 76 . In a horizontal tube like Fig. 76 a the diameters are 6 inches 
and 12 inches, and the heights of the pressure-columns or piezometers are 
12.16 feet and 12.96 feet above the same bench-mark. Find the loss of 
head between the two sections when the discharge is 1.57 cubic feet per 
second, and also when it is 4.71 cubic feet per second. 


Art. 77 . Loss Due to Contraction of Section 

When a sudden contraction of section in the direction of the 
flow occurs, as in Fig. 77 , the water suffers a contraction similar 
to that in the standard orifice, and hence in its expansion to fill 
the second section a loss of head results. Let V\ be the ve¬ 
locity in the larger section and v that 
in the smaller, while v' is the velocitv 
in the contracted section of the flowing 
stream; and let a\, a, and a’ be the 
corresponding areas of the cross-sec¬ 
tions. From the formula ( 76) 2 the loss 
of head due to the expansion of section 
from a' to a is 




7 77 ( 77 ), 

Fl S- 77 . \a J 22 \c / 22 


9 9 

zr 


,2 rjl 


,a j 2g \c / 2 g 

in which c' is the coefficient of contraction of the stream or the 
ratio of a' to a (Art. 44 ). 

The value of c' depends upon the ratio between the areas a 
and ai. When a is small compared with a\, the value of c' may 
be taken at 0.62 as for orifices (Art. 44 ). When a is equal to d\, 
there is no contraction or expansion of the stream and c' is unity. 
Let d and d\ be the diameters corresponding to the areas d and <21, 
and let r be the ratio of d to d\. Then experiments seem to in¬ 
dicate that an expression of the form 

n 


c' = m + 


1.1 — r 





























Loss Due to Contraction of Section. Art. 77 183 

gives the law of variation of c' with r. Placing c' = 0.62 and 
r — o gives one equation between m and n ; placing c' = 1.00 
and r = 1 gives another equation; and the solution of these fur¬ 
nishes the values of m and n. Thus is found 

c' = 0.582 + ^418 
1.1 — r 

from which approximate values of c' can ba computed : 

for r = 0.0 0.4 0.6 0.7 0.8 0.9 0.95 1.0 

c = 0.62 0.64 0.67 0.69 0.72 0.79 0.86 1.00 

from which intermediate values may often be taken without the 
necessity of using the formula. 

For a case of gradual contraction of section, such as shown in 
Fig. 7 5 a, the loss of head is less than that given by formula ( 77 ) 1, 
and it can only be determined from three measured quantities 
by the help of the general formulas of Art. 75 . If the change 
of section is made so that the stream has no subsequent enlarge¬ 
ment, loss of head is avoided; for, as the above discussions show, 
it is the loss in velocity due to sudden expansion which causes the 
loss of head. 

The loss due to sudden contraction of a tube or pipe is often 
much smaller than that due to sudden expansion. For instance, let 
the diameter of the large section be three times that of the smaller, 
and the velocity in the large section be 2 feet per second, then the loss 
of head which occurs when the flow passes from the small to the 
large section is, by Art. 76 , 

h' = 0.01555(18 — 2) 2 = 4.0 feet 

But if the flow occurs in the opposite direction, the ratio r is the co¬ 
efficient c' is about 0.64, and the loss of head is 

h r = 0.01555^—-i') 18 2 = 1.6 feet 

\0.64 J 

When, however, the ratio r is higher than 0.77, the loss due to sudden 
contraction is greater than that due to sudden expansion. Thus, if 
the diameter of the small section be nine-tenths that of the large one 




184 


Chap. 7 . Flow of Water through Tubes 


and the velocity in the large section be 2 feet per second, the loss of 
head when the flow passes from the small to the large section is 



But if the flow occurs in the opposite direction, the ratio r is 0.9, the 
coefficient c' is 0.79, and the loss of head is 



2.47 2 = 0.0066 feet 


As formula ( 77 ) 2 is an empirical one the results derived from it are tc 
be regarded as approximate. 

Prob. 77. Compute the loss of head when a pipe which discharges 1.57 


cubic feet per second suddenly diminishes in section from 12 to 6 inches in 
diameter. 


Art. 78 . The Standard Short Tube 


An adjutage is a tube inserted into an orifice, and the short- 
tube adjutage, consisting of a cylinder whose length is about 
three times its diameter, is the most common form. For 
convenience it will be called the standard short tube, because 


its theory and coefficients form a starting-point with which all 


other adjutages may be compared. This short tube is of little 
value for the measurement of water, since the coefficients for 
standard orifices are much more definitely known. The discus¬ 
sion here given is for the case where the inner edge is a sharp, 
definite corner like that of the standard orifice (Art. 43 ). When 
the tube is only two diameters in length, the stream passes through 


without touching it, as in 
Fig. 78 u,and the discharge 
is the same as from the 
orifice. When it is length¬ 
ened sufficiently, the 
stream expands and fills 
the tube, as in Fig. 7 8 b, 



Fig. 78 a. 


Fig. 78 b. 


and the discharge is much increased. By observations on glass 
tubes it is seen that the stream usually contracts after leaving 
the inner end of the tube and then expands. This contraction 









The Standard Short Tube. Art. 78 


185 


may be apparently destroyed by agitating the water or by strik¬ 
ing the tube, and the entire tube is then filled, yet if a hole is 
bored in the tube near its inner end, water does not flow out, but 
air enters, showing that a negative pressure exists. 

An estimate of the velocity and discharge from this short- 
tube adjutage may be made as follows: Let h be the head on the 
inner end of the tube and v the velocity of the outflowing water. 
The head h equals the velocity-head d 2 / 2g plus all the losses of 
head. At the inner edge a loss of o.n v 2 /2g occurs in entering 
the tube, as in the standard orifice (Art. 56 ), and then there is 
a loss of (v' — v) 2 /2g when the contracted stream suddenly ex¬ 
pands so that its velocity v' is reduced to v (Art. 76 ). If a' 
and a are the areas of these two sections, their ratio a'/a is the 
coefficient of contraction c'. Then 


h = o.n — 
2g 

4 



sr v* 

2g 2 g 


Now, taking for c' its mean value 0.62, this equation reduces 
to v = 0.82 V2g/z, or the coefficient of velocity of the issuing 
jet is 0.82. Since the cross-section of the stream at the outer 
end of the tube is the same as that of the tube, the coefficient 
of contraction for that end is unity, and hence (Art. 46 ) the mean 
value of the coefficient of discharge is also 0.82. 


While this theoretic discussion does not take account of losses 
due to the small frictional resistances along the sides of the tube 
after the stream has expanded, the mean results of the experi¬ 
ments of Venturi and Bossut give closely the same coefficient. 
Hence both theory and practice agree in establishing as an aver¬ 
age value for the short tube, 

Coefficient of discharge c = 0.82 


This coefficient, however, ranges from 0.83 for low heads to 0.79 
for high heads. It is greater for large tubes than for small ones, 
its law of variation being probably the same as for orifices (Art. 
47), but sufficient experiments have not been made to state defi¬ 
nite values in the form of a table. 


18 G 


Chap. 7 . Flow of Water through Tubes 


~i— m* 


‘ ' 




A standard orifice gives on the average about 61 percent of 
the theoretic discharge, but by the addition of a tube this may 
be increased to 82 percent. The velocity-head of the jet from 
the tube is, however, much less than that from the orifice. For, 
let v be the velocity and h the head, then (Art. 45 ) for the standard 

orifice z; = 0.98 V2 gh or v 2 /2g = 0.96 h 

and similarly for the standard tube 

v = 0.82 V2 gh or v 2 / 2 g = 0.67 h 

Accordingly the velocity-head of the stream from the standard 
orifice is 96 percent of the theoretic velocity-head, and that of 
the stream from the standard tube is only 67 percent. Or 
if jets are directed vertically upward from a standard orifice and 
tube, as in Fig. 78 c, that from the former rises to the height 0.96 //, 

while that from the latter rises 
to the height 0.67 h, where li is 
the head measured downward 
from the surface of water in 
the reservoir to the point of 
exit from the orifice. 

The energy lost in the 
stream from the standard ori¬ 
fice is hence 4 percent of the 
theoretic energy, but 33 per¬ 
cent is lost in the stream from 
the standard tube. In reality energy is never lost, but is merely 
transformed into other forms of energy. In the tube the one- 
third of the total energy which has been called lost is only lost 
because it cannot be utilized as work; it is, in fact, transformed 
into heat, which raises the temperature of the water. The above 
explanation shows that most of this loss is due to impact re¬ 
sulting from sudden expansion of the stream. 

The loss of head in the flow from the short tube is large, but 
not so large as might be expected from theoretical considerations 
based on the known coefficients for orifices. When the tube has 
a length of only two diameters, the water does not touch its 



n 








0.67 h 


Fig. 78c. 
































187 


The Standard Short Tube. Art. 78 

inner surface, and the flow occurs as from a standard orifice. 
The \ elocity in the plane of the inner end is then 61 percent of 
the theoretic velocity, since the mean coefficient of discharge 
is 0.61. Now when the tube is sufficiently increased in length, 
its outer end will be filled, and if the contraction still exists, it 
might be inferred that the coefficient for that end would be 
also 0.61 ; this would give a velocity-head of (0.61) 2 h or 0.37 /z, 
so that the loss of head would be 0.63 h. Actually, however, the 
coefficient is found to be 0.82 and the loss of head only 0.33 h. 
It hence appears that further explanation is needed to account 
for the increased discharge and energy. 

In the first place, a loss of about 0.04 h occurs at the inner end 
of the tube in the same manner as in the standard orifice, and only 
the head 0.96 h is then available for the subsequent phenomena. 
If the coefficient c' for the contracted section has the value 0.62, 
the velocity in that section is 

v' = V2 g/z = 1.3 2 V 2 gh 

0.62 

and the velocity-head for that section is 

v' 2 /2g = 1.75 h 

and consequently the pressure-head in that section is 

0.96 h — 1.75 h — — 0.79 h 

There exists therefore a negative pressure or partial vacuum near 
the inner end of the tube which is sufficient to lift a column of 
water to a height of about three-fourths the head. This conclu¬ 
sion has been confirmed by experiment for low heads, and was 
in fact first discovered experimentally by Venturi. For high 
heads it is not valid, since in no event can atmospheric pressure 
raise a column of water higher than about 34 feet (Art. 4 ); prob¬ 
ably under high heads the coefficient of contraction of the stream 
in the tube becomes much greater than 0.62. 

The cause of the increased discharge of the tube over the 
orifice is hence a partial vacuum, which causes a portion of the 
atmospheric head of 34 feet to be added to the head h, so that the 





188 


Chap. 7 . Flow of Water through Tubes 


flow at the contracted section occurs as if under the head h + h- 
The occurrence of this partial vacuum is attributed to the fric¬ 
tion of the water on the air. When the flow begins, the stream 

is surrounded by air of the normal at¬ 
mospheric pressure which is imprisoned 
as the stream fills the tube. The friction 
of the moving water carries some of this 
air out with it, thus rarefying the re¬ 
maining air. This rarefaction, or nega¬ 
tive pressure, is followed by an increased 
velocity of flow, and the process con¬ 
tinues until the air around the contracted 
section is so rarefied that no more is re¬ 
moved, and the flow then remains per¬ 
manent, giving the results ascertained 
by experiment. The partial vacuum causes neither a gain nor 
loss of head, for although it increases the velocity-head at the 
contracted section to 1.75 //, there must be expended 0.79 li in 
order to overcome the atmospheric pressure at the outer end of 
the tube. The experiments of Buff have proved that in an 
almost complete vacuum the discharge of the tube is but little 
greater than that of the orifice.* 

Prob. 78 . When the coefficient of contraction for the contracted sec¬ 
tion is 0.70, find the probable coefficient of discharge and also the negative 
pressure-head. 

Art. 79 . Conical Converging Tubes 

Conical converging tubes are used when it is desired to obtain 
a high efficiency in the energy of the stream of water. At A, Fig. 
79 , is shown a simple converging tube, consisting of a frustum of 
a cone, and at B is a similar frustum provided with a cylindrical 
tip. The proportions of these converging tubes, or mouthpieces, 
vary somewhat in practice, but the cylindrical tip when em¬ 
ployed is of a length equal to about 2\ times its inner diameter, 
while the conical part is eight or ten times the length of that 

* Annalen der Physik und Chemik, 1839, vol. 46, p. 242. 








































Conical Converging Tubes. Art. 79 


189 


diameter, the angle at the vertex of the cone being between 
io and 20 degrees. 


The stream from a conical converging tube like A suffers 
a contraction at some distance beyond the end. The coefficient 
of discharge is higher than 
that of the standard tube, 
being generally between 0.85 
and 0.95, while the coefficient 
of velocity is higher still. 

Experiments made by d’Au- 
buisson and Castel on conical 
converging tubes 0.04 meters long and 0.0155 meters in di¬ 
ameter at the small end, under a head of 3 meters, furnish 
the coefficients of discharge and velocity given in Table 79 . 



Fig. 79 . 


Table 79 . Coefficients for Conical Tubes 


Angle of Cone 

Discharge 

c 

V elocity 

Contraction 

c ' 

o° 

00' 

0.829 

0.829 

1.00 

1 

36 

.866 

.867 


4 

10 

.912 

.910 


7 

52 

•930 

•932 

0.998 

10 

20 

•938 

•951 

.986 

13 

24 

.946 

•963 

•983 

16 

36 

•938 

.971 

.966 

21 

00 

.919 

.972 

•945 

29 

58 

.895 

•975 

.918 

48 

50 

.847 

.984 

.861 


The former of these was determined by measuring the actual dis¬ 
charge (Art. 46 ), and the latter by the range of the jet (Art. 45 ). 
The coefficient of contraction as computed from these is given in the 
last column, and this applies to the jet at the smallest section, 
some distance beyond the end of the tube. While these values 
show that the greatest discharge occurred for an angle of about 
13J 0 , they also indicate that the coefficient of velocity in¬ 
creases with the convergence of the cone, becoming about equal 
to that of a standard orifice for the last value. Hence the table 






























190 


Chap. 7. Flow of Water through Tubes 


seems to teach that a conical frustum does not usually give as 
high a velocity as a standard orifice. 

Under very high heads, over 300 feet, Hamilton Smith found 
the actual discharge to agree closely with the theoretical, or the 
coefficient of discharge was nearly 1.0, and in some cases slightly 
greater.* His tubes were about 0.9 feet long, 0.1 feet in diameter 
at the small end and 0.35 feet at the large end, the angle of 
convergence being 17 0 . As these figures indicate a contrac¬ 
tion of the jet beyond the end, it cannot be supposed that the 
coefficient of discharge in any case was really as high as his ex¬ 
periments indicate. Under these high heads the cylindrical 
tip applied to the end of a tube produced no effect on the dis¬ 
charge, the jet passing through without touching its surface. 

Prob. 79 . When the coefficient of discharge of a tube is 0.98 and the 
coefficient of velocity of the jet is 0.995, compute the coefficient of contrac¬ 
tion of the jet. 


Art. 80 . Inward Projecting Tubes 

Inward projecting tubes, as a rule, give a less discharge than 
those whose ends are flush with the side of the reservoir, due to 
the greater convergence of the lines of direction of the filaments 
of water. At A and B , Fig. 80 , are shown inward projecting 
tubes so short that the water merely touches their inner edges, 
and hence they may more properly be called orifices. Experi¬ 
ment shows that the case at A, where the sides of the tube are 



normal to the side of the reservoir, gives the minimum coefficient 
of discharge c = 0.5, while for B the value lies between 0.5 and 
that for the standard orifice at C. The inward projecting cylin¬ 
drical tube at D has been found to give a discharge of about 
72 percent of the theoretic discharge, while the standard tube 


* Hydraulics (London and New York, 1886), p. 286. 


















Diverging and Compound Tubes. Art. 81 


191 


(Art. 78 ) gives 82 percent. For the tubes E and F the coefficients 
depend upon the amount of inward projection, and they are 
much larger than 0.72 for both cases, when computed for the 
area of the smaller end. 

It is usually more convenient to allow a water-main to pro¬ 
ject inward into the reservoir than to arrange it with its mouth 
flush to a vertical side. The case D , in Fig. 80 , is therefore of 
practical importance in considering the entrance of water into 
the main. As the end of such a main has a flange, forming a 
partial bell-shaped mouth, the value of c is probably higher 
than 0.72. The usual value taken is 0.82, or the same as for the 
standard tube. Practically, as will be seen later, it makes little 
difference which of these is used, as the velocity in a water-main 
is slow and the resistance at the mouth is very small compared 
with the frictional resistances along its length. 

Prob. 80 . Find the coefficient of discharge for a tube whose diameter 
is one inch when the flow under a head of 9 feet is 22.1 cubic feet in 3 minutes 
and 30 seconds. 


Art. 81 . Diverging and Compound Tubes 

In Fig. 81 is shown a diverging conical tube, BC, and two 
compound tubes. The compound tube ABC consists of two 
cones, the converging one, AB, be¬ 
ing much shorter than the diverg¬ 
ing one, BC, so that the shape 
roughly approximates to the form 
of the contracted jet which issues 
from an orifice in a thin plate. 

In the tube AE the curved con¬ 
verging part A B closely imitates 
the contracted jet, and BB is a 
short cylinder in which all the 
filaments of the stream are sup¬ 
posed to move in lines parallel to 
the axis of the tube, the remaining part being a frustum of a 
cone. The converging part of a compound tube is often called 
a mouthpiece and the diverging part an adjutage. 



Fig. 81 . 
























192 


Chap. 7. Flow of Water through Tubes 


Many experiments with these tubes have shown the inter¬ 
esting fact that the discharge and the velocity through the small¬ 
est section, B , are greater than those due to the head; or, in 
other words, that the coefficients of discharge and velocity for 
this section are greater than unity. One of the first to notice 
this was Bernouilli in 1738, who found c = 1.08 for a diverging 
tube. Venturi in 1791 experimented on such tubes, and showed 
that the angle of the diverging part, as also its length, greatly 
influenced the discharge. He concluded that c would have a 
maximum value of 1.46 when the length of the diverging part 
was nine times its least diameter, the angle at the vertex of the 
cone being 5 0 06'. Eytelwein found c = 1.18 for a diverging 
tube like BC in Fig. 81 , but when this tube was used as an ad¬ 
jutage to a mouthpiece AB, thus forming a compound tube ABC, 
he found c = 1.55. 

The experiments of Francis in 1854 on a compound tube like 
ABCDE are very interesting.* The curve of the converging 
part AB was a cycloid, BB was a cylinder, and the diameters at 
A, B, C, D, and E were 1.4, 0.102, 0.145, 0.234, and 0.321 feet. 
The piece BB was 0.1 feet long, and the others each 1 foot; 
these were made to screw together, so that experiments could 
be made on different lengths. A sixth piece, EF, not shown in 
the figure, was also used, which was a prolongation of the diverg¬ 
ing cone, its largest diameter being 0.4085 feet. The tubes were 
cast iron, and quite smooth. The flow was measured with the 
tubes submerged, and the effective head varied from about 0.01 
to 1.5 feet. Excluding heads less than 0.1 feet, the following 
shows the range in value of the coefficients of discharge: 


. for tube AB , 
for tube AC, 
for tube AD, 
for tube AE, 
for tube AF, 


c for Section BB 

0.80 to 0.94 
1.43 to 1.59 
1.98 to 2.16 
2.08 to 2.43 
2.05 to 2.42 


c for Outer End 

0.80 to 0.94 
0.70 to 0.78 
0.37 to 0.41 
0.21 tO O.24 
0.13 to 0.15 


* Lowell Hydraulic Experiments, 4th Edition, pp. 209-232. 


Diverging and Compound Tubes. Art. 81 


193 


The maximum discharge was thus found to occur with the tube 
AE, and to be 2.43 times the theoretic discharge that would be 
expected for the small section BB. In general the coefficients 
increased with the heads, the value 2.08 being for a head of 0.13 
feet and 2.43 for a head of 1.36 feet; for 1.39 feet, however, c 
was found to be 2.26. 

These coefficients of discharge are the same as the coefficients 
of velocity, since the tube was entirely filled. Thus, when the 
coefficient for the section BB was 2.43, the velocity was 

v = 2.43 V 2gh, 
and the velocity-head was 

v 2 /2g= (2.43) 2 // = 5.90/; 

Therefore the flow through the section BB was that due to a head 
5.9 times greater than the actual head of 1.36 feet; or, in other 
words, the energy of the water flowing in BB was 5.9 times the 
theoretic energy. Here, apparently, is a striking contradic¬ 
tion of the fundamental law of the conservation of energy. 
The explanation of this apparent contradiction is the same as 
that given in Art. 78 for the short-tube adjutage. The increased 
velocity and discharge is due to the occurrence of a partial vac¬ 
uum near the inner end of the adjutage BC. The pressure of the 
atmosphere on the water in the reservoir thus increases the hydro¬ 
static pressure due to the head, and the increased flow results. 
The energy at the smallest section is accordingly higher than the 
theoretic energy, but the excess of this above that due to the head 
must be expended in overcoming the atmospheric pressure on 
the outer end of the tube, so that in no case does the available 
exceed the theoretic energy. No contradiction of the law of 
conservation therefore exists. 

To render this explanation more definite, let the extreme case be 
considered where a complete vacuum exists near the inner end of the 
adjutage, if that were possible, as it perhaps might be with a tube of 
a certain form. Let h be the head of water in feet on the center of 
the smallest section. The mean atmospheric pressure on the water 
in .the reservoir is equivalent to a head of 34 feet (Art. 4 ). Hence 
the total head which causes the discharge into the vacuum is h -f- 34 



194 


Chap. 7. Flow of Water through Tubes 


and the velocity of flow is nearly V2g(/*-f 34). Neglecting the re¬ 
sistances, which are very slight if the entrance is curved, the coefficients 
of velocity and discharge can now be found ; thus : 

for h = 100, z> = V2g X 134 = 1.16 V2 gh 

iorh= 10, v — ~s/2g X 44 = 2.10V2 gh 

for h = 1, v = V 2 gX 35 = 5.92V2 gh 

The coefficient hence increases as the head decreases. That this is 

not the case in the above experiments is undoubtedly due to the fact 
that the vacuum was only partial, and that the degree of rarefaction 
varied with the velocity. The cause of the vacuum, in fact, is to be 
attributed to the velocity of the stream, which by friction removes a 
part of the air from the inner end of the adjutage. 

It follows from this explanation that the phenomena of increased 
discharge from a compound tube could not be produced in the absence 
of air. The experiment has been tried on a small scale under the re¬ 
ceiver of an air-pump, and it was found that the actual flow through 
the narrow section diminished the more complete the rarefaction. 
It also follows that it is useless to state any value as representing, 
even approximately, the coefficient of discharge for such tubes. 

Prob. 81 . Compute the pressure per square inch in the section BB of 
Francis’ tube when h = 1.36 feet and c = 2.43. What is the height of the 
column of water that can be lifted by a small pipe inserted at BB ? 

Art. 82 . Submerged Tubes 

As shown in Art. 51 the effective head h which causes the flow 
through a submerged orifice or tube is the difference in the level 

of the water above and below 
the orifice or tube. This dif¬ 
ference h , as in Fig. 82 , also 
represents the loss of head oc¬ 
casioned by the flow through 
the tube. The discharge 
through a submerged tube is 
probably somewhat less than that from the same tube when dis¬ 
charging freely into the air. Stewart,* at the laboratory of the 



* Engineering Record, Sept. 28, 1907. 






























Submerged Tubes. Art. 82 


195 


University of Wisconsin, experimented on large submerged tubes 
from 4 feet by 4 feet square. These tubes varied in length 
from 0.3 to 14.0 feet, while the heads h ranged from 0.05 to 
0.30 feet. Experiments were made under various conditions of 
entrance by placing at the mouth of the tubes an elliptical 
mouthpiece as shown in Fig. 82 . This mouthpiece was made 
in four parts, and after experiments with the straight square- 
edged tube had been run, others with the bottom of the mouth¬ 
piece in place, with the bottom and one side, with the bottom and 
two sides, and with all four of its parts in position were made. 

In the following table are shown the results of these experi¬ 
ments; the coefficients in the first line opposite each head being 
those for the square-edged tube, while those in the second line 
are for the same tube with the full elliptical mouthpiece in posi¬ 
tion as shown. 

Table 82 . Coefficients for Submerged Tubes 


Head 




Length of Tube in 

Feet 



in 

Feet 


0.31 

0.62 

125 

2.50 

i 

5.00 

10.00 

14.00 

0.05 

/ 

0.631 

0.650 

0.672 

0.769 

0.807 

0.824 

0.838 

i 

.948 


•943 

.940 

.927 

•93i 


/ 

o.6n 

0.631 

0.647 

0.718 

0.763 

0.780 

0-795 


t 

•932 



.911 

.899 

.892 

•893 


/ 

0.609 

0.628 

0.644 

0.708 

0.758 

0.779 

0.794 

0.15 

l 

•936 



.910 

.899 

•893 

.894 

0.20 

{ 

0.609 

.948 

0.630 

0.647 

0.711 

•923 

0.768 

.911 

0.794 

.906 

0.809 

•905 


/ 

0.610 

0.634 

0.652 

0.720 

0.782 

0.812 

0.828 

0.25 

l 

•965 



•938 

.928 



0.30 


0.614 

0.639 

0.660 

0-731 

0.769 

0.832 

0.850 


From an inspection of these results it appears that the coeffi¬ 
cients for the square-edged tubes increase both with the head and 
with the length of the tube, while for the tubes fitted with the 
mouthpiece they increase with the head but decrease with the 
length of the tube. This behavior is readily explained if it 
be remembered that the larger quantities carried with the mouth¬ 
piece in position must cause more friction and so cause a reduction 






















196 


Chap. 7. Flow of Water through Tubes 


in the effective head. The length of the square-edged tubes 
experimented on was evidently not sufficient to cause the friction 
in them to overcome the tendency to greater discharge due to 
contraction at entrance and subsequent expansion in the tube. 

Prob. 82 . What will be the discharge through a submerged square- 
edged tube 5 feet by 4 feet in section and 10 feet long, when the difference 
between the water levels above and below it is 0.5 feet? 

Art. 83 . Nozzles and Jets 

For fire service two forms of nozzles are in use. The smooth 
nozzle is essentially a conical tube like A in Fig. 79 , the larger 
end being attached to a hose, but it is often provided with a cylin¬ 
drical tip and sometimes the larger end is curved, as shown in 
Fig. 83 a. The ring nozzle is a similar tube, but its end is con- 



Fig. 83a. Fig. 83 b. 


tracted so that the water issues through an orifice smaller than 
the end of the tube. The experiments of Freeman show that the 
mean coefficient of discharge is about 0.97 for the smooth nozzle 
and about 0.74 for the ring nozzle* The smooth nozzle is used 
much more than the ring nozzle. 

Let d be the diameter of the pipe or hose and D the diameter 
of the outlet at the end of the nozzle, and let v and V be the cor¬ 
responding velocities. Let h\ be the pressure-head at the en¬ 
trance to the nozzle; then the effective head at the entrance to 
the nozzle is , 

= /*i + - 

2 g 

and the velocity at the end of the nozzle is V = d V^gH, where 
Ci is the coefficient of velocity. The reasoning of Art. 50 applies 
here, if the ratio D 2 /d 2 is used in place of a/A, and hi in place 
of //, and hence 

V = ci 

Transactions American Society of Civil Engineers, 1889, vol. 21, pp. 303-482. 




i - c 2 (D/d ) 4 


( 83 ); 


















Nozzles and Jets. Art. 83 


197 


is the velocity of flow from the nozzle, c being the coefficient of 
discharge. The discharge per second is, from formula ( 50 ) 2 , 




(i A ) 2 - (D/dy 



The effective head at the nozzle entrance is 


i V 2 Jh 

Ci 2 2 g i —c 2 (D/d ) 4 
and the velocity-head of the issuing jet is 

F 2 _ d 2 hi 
2g i — c 2 {D/d) 

which gives the height to which the jet would rise if there were 
no atmospheric resistances. In these formulas D/d is an ab¬ 
stract number, and to find its value D and d may be taken in any 
unit of measure. 


When hi and D are in feet, g is to be taken as 32.16 feet per 
second per second. Then ( 83 ) 1 gives F in feet per second and 
( 83) 2 gives q in cubic feet per second. When the gage at the nozzle 
entrance gives the pressure pi in pounds per square inch, hi in 
feet is found from 2.304 pi. It is a common practice in figuring 
on fire-streams to compute the discharge in gallons per minute. 
For this case, if D is taken in inches, 

9 = -\ D /d)* 

gives the discharge in gallons per minute. 

For smooth nozzles the value of the coefficient of velocity 
Ci is the same as that of the coefficient of discharge c, since the jet 
issues without contraction. The experiments of Freeman fur¬ 
nish the following mean values of the coefficient of discharge for 
smooth cone nozzles of different diameters under pressure-heads 
ranging from 45 to 180 feet: 

Diameter in inches = f f 1 i| 1} if 

Coefficients = 0.983 0.982 0.972 0.976 0.971 0.959 

These values were determined by measuring the pressure pi 
and the discharge q, from which c can be computed by the last 








198 Chap. 7 . Flow of Water through Tubes 

formula. For example, a nozzle having a diameter of i.ooi 
inches at the end and 2.50 inches at the base discharged 208.5 
gallons per minute under a pressure of 50 pounds per square 
inch at the entrance. Here D = 1.001, d = 2.5, pi = 50, and 
q = 208.5, an d inserting these in the formula and solving for 
c } there is found c = 0.985. 

In ring nozzles the ring which contracts the entrance is usually 
only yq or § inch in width. The effect of this is to diminish the 
discharge, but the stream is sometimes thrown to a slightly greater 
height. On the whole, ring nozzles seem to have no advantage 
over smooth ones for fire purposes. As the stream contracts 
after leaving the nozzle, the coefficient of velocity Ci is greater 
than the coefficient of discharge c. The value of c being about 
0.74, that of Ci is probably a little larger than 0.97. In using 
( 83 ) 1 for ring nozzles these values of C\ and c should be inserted, 
but in using ( 83) 2 only the value of c is needed. 

According to Freeman’s experiments, the discharge of a 
J-inch ring nozzle is the same as that of a f-inch smooth nozzle, 
while the discharge of a 1 J-inch ring nozzle is about 20 percent 
greater than that of a i-inch smooth nozzle. The heights of 
vertical jets from a 1 J-inch ring nozzle are about the same as those 
from a i-inch smooth nozzle, while the jets from a if-inch 
ring nozzle are slightly less in height than those from a ij-inch 
smooth nozzle. 

The vertical height of a jet from a nozzle is very much less, 
on account of the resistance of the air, than the value deduced 
above for F 2 /2 g. For instance, let a smooth nozzle 1 inch in 
diameter attached to a 2.5-inch hose have c = 0.97 and the pres¬ 
sure-head hi = 230 feet; then the computation gives the velocity- 
head V 2 /2g as 221 feet, whereas the average of the highest drops 
in still air will be about 152 feet high and the main body of water 
will be several feet lower. Table 83 , compiled from the results 
of Freeman’s experiments, shows for three different smooth 
nozzles the height of vertical jets, column A giving the heights 
reached by the average of the highest drops in still air, and column 
B the maximum limits of height as a good effective fire-stream 


Nozzles and Jets. Art. 83 


199 


Table 83 . Vertical Jets from Smooth Nozzles 

* 


Indicated 
Pressure at 
Entrance 
to Nozzle 
Pounds per 
Square 
Inch 

From 2 -inch Nozzle 

From 1 -inch Nozzle 

From 1 2 -inch Nozzle 

Height in Feet 

Dis¬ 

charge 

Gallons 

per 

Minute 

Height in Feet 

Dis¬ 

charge 

Gallons 

per 

Minute 

Height in Feet 

Dis¬ 

charge 

Gallons 

per 

Minute 

A 

B 

A 

B 

A 

B 

IO 

20 

17 

52 

21 

18 

93 

22 

19 

148 

20 

40 

33 

73 

43 

35 

J 32 

44 

37 

209 

30 

59 

48 

90 

63 

5 i 

161 

66 

53 

256 

40 

78 

60 

104 

83 

64 

186 

86 

67 

296 

50 

93 

67 

116 

101 

73 

208 

107 

77 

331 

60 

104 

72 

127 

117 

79 

228 

126 

85 

363 

70 

114 

76 

137 

130 

85 

246 

140 

91 

392 

80 

123 

79 

147 

140 

89 

263 

150 

95 

419 

90 

129 

81 

156 

147 

92 

279 

157 

99 

444 

IOO 

134 

83 

164 

152 

96 

295 

161 

101 

468 


with moderate wind. The discharges given depend only on the 
pressure, and are the same for horizontal as for vertical jets. 

The maximum horizontal distance to which a jet can be thrown 
is also a measure of the efficiency of a nozzle. The following, 
taken from Freeman’s tables, gives the horizontal distances at 
the level of the nozzle reached by the average of the extreme 
drops in still air. The practical horizontal distance for an effective 
fire-stream is, however, only about one-half of these figures. 

Pressure at nozzle entrance, 20 40 60 80 100 pounds. 

From f-inch smooth nozzle, 72 112 136 153 167 feet. 

From i-inch smooth nozzle, 77 133 167 189 205 feet. 

From ij-inch smooth nozzle, 83 148 186 213 236 feet. 

From ij-inch ring nozzle, 76 131 164 186 202 feet. 

From ij-inch ring nozzle, 78 138 172 196 215 feet. 

From if-inch ring nozzle, 79 144 180 206 227 feet. 

The ball nozzle, often used for sprinkling, has a cup at the end of 
the nozzle and within the cup a ball, so that the jet issuing from the 
tip of the nozzle is deflected sidewise in all directions. This apparatus 
exhibits a striking illustration of the principle of negative pressure, 
for the ball is not driven away from the tip, but is held close to it by 
the atmospheric pressure, the negative pressure-head being caused by 
































200 


Chap. 7. Flow of Water through Tubes 


the high velocity of the sheet of water around the ball. The cup is 
usually so arranged that the ball cannot be driven out of it, for this 
might occur under the first impact of the jet, but when the flow has 
become steady, there is no tendency of this kind, and the ball is seen 
slowly revolving upon the cushion of water without touching any part 
of the cup. 

Prob. 83 . A nozzle if inches in diameter attached to a play-pipe 2\ 
inches in diameter discharges 310.6 gallons per minute under an indicated 
pressure of 30 pounds per square inch. Find the velocity of the jet and the 
coefficient C\. 

Art. 84. Lost Head in Long Tubes 


When water issues from an orifice, tube, pipe, or nozzle 
with the velocity v, its velocity-head is v 2 /2g, and it is only this 
part of the total effective head h that can be utilized for the pro¬ 
duction of work. The lost head then is 

»,2 

h =h~ — 

2 g 

Now if Ci is the coefficient of velocity for the section where the 
discharge occurs, the velocity v is given by C\ 2gh, and hence 


h'= 


I I — 


(84), 


/ 2 g 

is a general expression for the lost head in terms of the velocity- 
head. For the standard orifice (Art. 45 ), the mean value of C\ 
is 0.98 and for an orifice perfectly smooth c\ is 1.00; hence 
from ( 84 ), 


v 2 


and ti = o 


ti = 0.04 — 
are the losses of head for these two cases 


For the standard short cylindrical tube (Art. 78) the value 
of C\ is about 0.82, and the loss of head is 


ti = 


\ zr zr 

- 1 )— = 0.49 — 

J 2g 2g 


nO-82 2 

For the inward projecting cylindrical tube (Art. 80) the value 
of Ci is about 0.72, and hence the loss of head is 

,2 


ti = 


^0.72^ 


_i>! 

)2g 


o -93 


LI 





Lost Head in Long Tubes. Art. 84 201 

Accordingly the loss of head for the inward projecting tube is 
nearly equal to the velocity-head of the issuing stream, while 
that from the standard tube is about one-half the velocity-head. 

When a tube is longer than three diameters, it becomes a long 
tube or a pipe. Here the loss of head is much greater because 
the water meets with frictional resistances along the interior sur¬ 
face, and the longer the pipe, the greater is this resistance and the 
slower is the velocity. The formula ( 84 )i gives the total loss of 
head for this case also. For example, the experiments of Eytel- 
wein and others have given values of c\ for the cases below, and 
from these the corresponding values of the total lost head have 
been computed. Let l denote the length of the pipe and d its 
diameter, the end connected with the reservoir being arranged 
like the standard tube; then 

for/=i2cf ci = 0.77 h'= o.6gv 2 /2g 

for l = 3 6d C\ = 0.67 It = 1.23 v 2 /2g 

for l =■ 60 d Ci = 0.60 ti = 1.77 v 2 /2g 

Now in each of these cases the amount 0.49 v 2 /2g is lost in enter¬ 
ing the tube and in impact, as in the standard short tube. Hence 
the loss of head in friction in the remaining length of the pipe 
is h " = li' — 0.49V 2 /2g, or 

for l = 1 2d h" = 0.20 v 2 /2g 

for / = 36J h" = o. h ]4V 2 /2g 

for l = 60 d h" = 1.28 v 2 / 2g 

which shows that the frictional losses increase with the length 
of the pipe. The length of the pipe in which the entrance losses 
occur is about 3 d\ hence if 3 d be subtracted from each of the above 
lengths, the lengths in which the friction loss occurs are 9 d, 33 d, 
and 57 d, and it is seen that the above losses of head in friction 
are closely proportional to these lengths. By these and many 
other experiments it has been shdwn that the loss of head in 
friction varies directly with the length of the pipe. 

The lost head has here been expressed in terms of the velocity- 
head, but it can also be expressed in terms of the total head h 


202 


Chap. 7 . Flow of Water through Tubes 


that causes the flow. For, substituting in ( 84 )i the value of v 
given by C\'\/2gh, it reduces to 

ti — (i — ci 2 )// ( 84) 2 

Thus, for the standard short tube h' = 0.33 h ; for the inward 
projecting tube h' = 0.48 h , and for the above tube or pipe whose 
length is 60 diameters /z' = 0.64 h. 

Prob 84 . Find the ratio of the kinetic energy in the jet from a standard 
orifice to that in the jet from a standard tube, the diameters of orifice and 
tube being the same. 

Art. 85 . Inclined Tubes and Pipes 

The tubes discussed in this chapter have generally been re¬ 
garded as horizontal, but, if this is not the case, the formulas for 
velocity and discharge may be applied to them by measuring the 
head from the water level in the reservoir down to the center of 
the head of the pipe. Thus, for the nozzles of Art. 83 , it is under¬ 
stood that the tip is at the same level as the gage which registers 
the pressure pi or the pressure-head hi ; if the tip be lower than 
the gage by the vertical distance di, the true pressure-head to be 
used in the formula is hi + di\ if it be higher, the true pressure- 
head is h — d\. Then the velocity-head zz 2 /2 g is to be measured 
upward from the tip of the nozzle. 

The theorem of Bernouilli, given in Art. 31 , is true for inclined 
as well as for horizontal pipes under uniform flow, but it will be 

convenient to express it 
in a slightly different 
form. Let ai and a 2 be 
two sections of a pipe 
where the velocities are 
Vi and v 2 , and the pres- 
N sure-heads are hi and /z 2 , 
and let the flow be steady 
so that the same weight 
of water, W, passes each section in one second. Let MN be 
any horizontal plane lower than the lowest section, as for in¬ 
stance the sea level, and let ei and e 2 be the elevations of ai 












Inclined Tubes and Pipes. Art. 85 


203 


and #2 above it. With respect to this plane the weight W at a\ 
has the potential energy Wei, the pressure-energy Wh\, and the 
kinetic energy W • v^/2g, or the total energy is 

w(e l + h 1 + ^) 

\ 2 gJ 

Similarly with respect to this plane the energy of W in a 2 is 

wU + ht + £) 

\ 2 gJ 

If no losses of energy occur between the two sections, these 
expressions are equal, and hence 

£1 + + — = e 2 + hi + — ( 85 )i 

2 g 2 g 

and hence the theorem of Bernouilli may be stated as follows: 

In any pipe, under steady flow without impact or friction, the 
gravity-head plus the pressure-head plus the velocity-head is a con¬ 
stant quantity for every section. 

Now let II 1 and H 2 be the heights of the water levels in the piezom¬ 
eter tubes above the datum plane; then ei + hi = Hi and e 2 T h 2 
= H 2 , and accordingly ( 85 ) 1 becomes 

Hi + — = 11 * + — ( 85) 2 

or, the piezometer elevation for ai plus the velocity-head is equal 
to the sum of the corresponding quantities for any other section. 

This theorem belongs to theoretical hydraulics, in which 
frictional resistances are not considered. Under actual conditions 
there is always a loss of energy or head, so that when water flows 
from ai to a 2 , the first member of the above equation is larger than 
the second. Let Wh r be the loss in energy, then this is equal to 
the difference of the energies in ai and a 2 with respect to the 
datum plane, and 

h f = (ei + //]) — (e 2 + h 2 ) + —— -^L 

‘ 2 g 2 g 

h' = H l -H 2 + — -- 

2P 2( 


or 


(80)3 











204 


Chap. 7. Flow of Water through Tubes 


that is, the lost head is equal to the difference in level of the water 
surfaces in the piezometer tubes plus the differences of the veloc¬ 
ity-heads. When the pipe is of the same size at the two sections, 
the velocities Vi and v 2 are equal when the flow is uniform, and 
the lost head is simply 

b! = H l - H 2 ( 85) 4 

Piezometers or pressure gages hence furnish a very convenient 
method of determining the head lost in friction in a pipe of uni¬ 
form size. For a pipe of varying section the velocities i\ and v 2 
must also be known, in order to use ( 85)3 for finding the lost head. 

Prob. 85 . A large Venturi water meter placed in a pipe of 57.823 square 
feet cross-section had an area of 7.047 square feet at the throat. When 
the discharge was 54.02 cubic feet per second, the elevations of the water 
levels in the piezometers at a\ and a 2 in Fig. 38 a were 99.858 and 98.951 
feet. Compute the loss of head between the two sections. 


Art. 86. Velocities in a Cross-section 


Thus far the velocity has been regarded as uniform over the 
cross-section of the tube or pipe. On account of the roughness 
of the surface, however, the velocity along the surface is always 
smaller than that near the middle of the cross-section. There 
appears to be no theoretical method of finding the law which 
connects the velocity of a filament with its distance from the 
center of the pipe, and yet it is probable that such a law exists. 
The mean velocity is evidently greater than the velocity at the 
surface and less than the velocity at the middle, and if the position 
of a filament were known whose velocity is the same as the mean 

velocity, a Pitot tube (Art. 41 ) with its tip 
at that position would directly measure 
the mean velocity. 

Let Fig. 86a be a longitudinal section 
of a pipe, and let A B be laid off to repre¬ 
sent the surface velocity v s and CD to represent the central ve¬ 
locity v c . Then the velocity v at any distance y from the axis 
will be an abscissa parallel to the axis and limited by the line AC 
and the curve ED. Suppose this curve to be a parabola whose 


B 


. c 



Fig. 86 a. 






Velocities in a Cross-section. Art. 86 


205 


equation, is y~ = mx, the origin being at D and x measured 
toward the left. When y is equal to the radius of the pipe r, the 
x alue of x is v c — v 8 and hence m = r 2 /(y c — v 8 ). The velocity 
V v lh e distance y above the axis is v c — x, and accordingly 

v y = v c - ( v c - v s ) y 2 /r 2 (86)i 

It thus is seen that the velocity at any distance from the axis 
cannot be found unless the surface and central velocities are 
known. The position of the filament having the same velocity 
as the mean velocity v can, however, be determined, since the 
mean velocity is the mean length of the solid of revolution whose 
section is shown by the broken lines. This solid consists of a 
cylinder having the volume n rr 2 v 8 and a paraboloid having the 
volume Wr 2 (v c — v 8 ), and the sum of these is \n rr 2 (y c + v s ). Divid¬ 
ing this by the area of the cross-section gives §(z> c + v 8 ) as the 
value of the mean velocity, and inserting this for v y in the above 
equation there is found y = 0.71 r for the ordinate of a filament 
whose velocity is the same as mean velocity v. If the parabolic 
curve gives the true law of variation of velocity, a Pitot tube 
with its tip placed 0.29^ below the top of the pipe would measure 
the mean velocity directly. 

The first measurements of velocities of filaments were made by 
Freeman in 1888 with the Pitot tube.* They were on jets issuing 
from fire nozzles and also from a if-inch tube under high velocities. 
For smooth nozzles the velocities were practically constant for 
a distance of 0.6 r from the center, and then rapidly decreased, 
and the ratio of the surface velocity to the central velocity was 
about 0.77. For the pipe the velocities decreased quickly near 
the center, but more rapidly toward the surface. The velocity 
curve for the nozzle lies outside and that for the pipe lies within 
the parabolic curve represented by the equation (86)1. 

Bazin made experiments in 1893 on jets from standard ori¬ 
fices, using also the Pitot tube.f He found the velocities near the 
center to be smaller than others within 0.2 r of the surface. Thus 

* Transactions American Society of Civil Engineers, 1889, vol. 21, p. 412. 
f Experiments on the Contraction of the Liquid Vein. Trautwine’s 
translation. New York, 1896. 


206 


Chap. 7 . Flow of Water through Tubes 


if v y = c V2 gk, the following are some of his values of c for a ver¬ 
tical circular and a vertical square orifice, h being always the head 
on the center. 

r = + o.8 +o.6 +0.2 o.o —0.2 —0.6 —0.8 

c— o.68 0.64 0.62 0.63 0.64 0.72 0.86 

c— 0.71 0.67 0.64 0.64 0.65 0.71 0.82 

These are for velocities in the plane of the orifice, and he found 
similar variations for a section of the jet at a distance from the 
orifice of about one-half its diameter. 

Judd and King,* in their experiments on orifices (Art. 45 ), 
traversed the jets with a Pitot tube and found that at the con¬ 
tracted section the velocity in all parts of the jet was uniform. 

Cole, in 1897, made measurements of velocities in pipes,t 
using the Pitot tube with a differential gage (Art. 37 ). For 
pipes 4, 6, and 12 inches in diameter he found the ratio of the mean 
velocity to the center velocity to range from 0.91 to 1.01, while 
for a 16-inch pipe he found it to range from 0.83 to 0.86. His 
velocity curves show that the surface velocity was 60 percent or 
more of the center velocity. 

Williams, Hubbell, and Fenkell, in 1899, made numerous 
measurements of velocities in water mains with the Pitot tube, 
and arrived at the conclusions that the ratio of the mean velocity 
to the central velocity was about 0.84, and that the surface velocity 
was about one-half the central velocity.{ These ratios agree with 
an ellipse better than with a parabola. Let the curve BD in Fig. 
86a be an ellipse having the semi-axes ED and BE , the ellipse 
being tangent to the pipe surface at B. As before, let AB repre¬ 
sent the surface velocity v s and CD the central velocity v c ; then 
ED is v c — v s and BE is the radius r. The equation of the ellipse 
with respect to E as an origin is 

(v e — v,) 2 y 2 + r 2 * 2 = (*’c ~ v t ) 2 r 2 


* Engineering News, Sept. 27, 1906. 

f Transactions American Society of Civil Engineers, 1902, vol. 47, p. 276. 
t Transactions American Society of Civil Engineers, 1902, vol. 47, p. 63. 



Velocities in a Cross-section. Art. 86 


207 


in which * is measured toward the right and 3/ upward. The 
velocity v y at any distance y from the axis CD is v s + x, and 
accordingly ^+ (r . _ 5<) (86), 

Now the mean velocity is the mean length of the solid of revolu¬ 
tion formed by the cylinder whose volume is nrr 2 v s and the semi¬ 
ellipsoid whose volume is f 7 rr 2 (v c - v 8 ). The volume of the solid 
is hence 7rr 2 (§ v c + f v s ) and the mean velocity is f v c +1 v 8 . Insert¬ 
ing this for v y in (86)2, there is found y = 0.75 r for the position of 
the filament having the same velocity as the mean velocity, while 
the parabola gave y = o.jir. If v 8 is one-half of v c , the mean 
velocity under the elliptic law is § v c + \ v 8 = 0.83^, while under 
the parabolic law it is \v e + \v, = o.y$v e . 

Much irregularity is observed in velocity curves plotted from 
actual measurements, this being due to pulsations in the water 
and to errors of observations. The above experiments were 
on pipes having diameters of 12, 16, 30, and 42 inches and under 
velocities ranging from 0.5 to 7.5 feet per second; and they are 
a very valuable addition to the knowledge of this subject. The 
conclusion that v s is one-half of v c is, however, one that appears to 
be liable to some doubt. The conclusion that the mean velocity 
v is about 0.847^ appears well established, and a Pitot tube with 
its tip at the center of the pipe will hence determine a fair value 
of the mean velocity, several readings being taken in order to 
eliminate errors of observation. 



In the case of fountain flow (Art. 87 ), Lawrence and Braun- 
worth * found that the velocities in the cross-section depend on 
whether or not the flow out of the top of the pipe occurs as in a 

* Transactions American Society of Civil Engineers, vol. 57. 




























208 


Chap. 7 . Flow of Water through Tubes 


jet or as over a weir. Thus, in Fig. 86ft the velocity curves for a 
vertical 6-inch cast-iron pipe are shown for velocities ranging from 
2 to 17 feet per second. These velocities were obtained from the 
expression v = V2 gh, where h was measured by a Pitot tube. 

Prob. 86. Let v s = 3 and v c = 6 feet per second. Plot the parabola 
from formula (86)1 and the ellipse from formula (86)2. 

Art. 87 . Fountain Flow 

When a stream of water rises and flows out of the top of a 
vertical pipe of diameter D , the flow, if the head 11 to which it rises 
above the top of the pipe is small, is practically the same as that 
over a thin-edged circular weir. As H increases there comes a 
transition period during which the character of the flow resembles 
neither that over a circular weir nor that of a jet. Lawrence and 
Braunworth * experimented on the fountain flow of water from 
pipes 2, 4, 6, 9, and 12 inches in diameter. They measured the 
heads 11 both by means of a Pitot tube and by sighting on two 
rods and across the top of the pipe. The water discharged during 
the experiments was measured volumetrically. From the dis¬ 
cussion of these experiments the following formulas were deduced: 

q = 8.80 D l *H 1 '* and q = 5.57 D l "H 053 

the first being for weir and the second for jet flow. Here D and 11 
are in feet and q in cubic feet per second, 11 being measured by 
means of sighting across the top of the flow as above described. 

For cases in which the head 11 is measured with a Pitot tube 
the formulas deduced were 

q = 8.80 D l2 »H i- 29 and q = 5.84 Z) 2025 # 0 - 53 

the first of these, as before, being applicable to weir and the sec¬ 
ond to jet flow. 

In general the average results given by these formulas are 
correct within 3 percent for the jet condition, while for the condi¬ 
tion of the weir flow using the Pitot tube for the measurement 
of the head the average accuracy is within 4 percent. Single 


* Transactions American Society of Civil Engineers, vol. 57, p. 209. 



Fountain Flow. Art. 87 


209 


measurements cannot be depended upon closer than to within 
about twice the above limits of accuracy. 

In the following table are shown the computed discharges in 
cubic feet per second for various sizes of pipes under various 
heads, the heads being observed by means of a Pitot tube. 


Table 87 . Discharges in Cubic Feet per Second for 
Fountain Flow from Vertical Pipes 


Head 



Diameter of Pipe in Inches 



in 

Feet 

1 

2 

4 

6 

8 

12 

18 

24 

0.02 

O.04 


0.014 

0.014 

O.035 

0.023 

0.055 

0.033 

0.080 

0.055 

0.133 

0.092 

0.223 

0.134 

0.324 

O.06 


0.023 

O.059 

0.093 

0.136 

0.227 

0.380 

0-549 

0.08 

0.010 

0.032 

0.085 

0.136 

0.197 

0.330 

0-550 

0.802 

0.10 

O.OII 

0.039 

0.II4 

0.180 

0.262 

0-439 

0 - 73 1 

1.08 

0.15 

0.014 

0.054 

O.184 

0.307 

0.442 

0.742 

1.28 

1.84 

0.20 

0.016 

0.065 

O.243 

0.438 

0.645 

1.08 

1.87 

2.66 

O.3O 

0.020 

0.082 

0.325 

0.662 

1.03 

1.81 

3.12 

4-45 

O.4O 

0.023 

0.096 

O.385 

0.832 

1.36 

2.66 

4-50 

6.50 

O.50 

0.026 

0.108 

0-435 

0.975 

1.65 

3-35 

5-98 

8.62 

o -75 

0.033 

0.133 

0-539 

1.23 

2.18 

4-73 

9 - 5 o 

14.80 

1.00 

O.O38 

0.155 

O.627 

1-43 

2.57 

5-73 

12.27 

20.20 

1.50 

O.O47 

0.192 

O.778 

1.77 

3.18 

7.22 

16.25 

28.08 

2.00 

0.055 

0.224 

0.906 

. 2.06 

3 - 7 i 

8.41 

I 9 -I 5 

33-75 

3.00 

0.068 

0.278 

1.16 

2.56 

4.60 

10.42 

23.80 

42.55 

4.00 

0.079 

0.324 

I.32 

2.98 

5-36 

12.15 

27.70 

49.60 

•5.00 

O.089 

0.365 

I.47 

3-3 6 

6.03 

13-67 

31.20 

55 - 8 o 

6.00 

O.098 

0.401 

I.62 

3 - 7 o 

6.64 

15-05 

34-40 

61.40 

7.00 

0.107 

0-435 

I.76 

4.02 

7.20 

16.34 

37-30 

66.70 

8.00 

O.II5 

0.467 

I.89 

4 - 3 1 

7-73 

17-55 

40.05 

71.60 

9.00 

0.122 

0.498 

2.01 

4-59 

8.23 

18.66 

42.65 

76.20 

10.00 

0.129 

0.527 

2.13 

4.86 

8.70 

19.79 

45.!0 

80.55 


In the above table the condition of weir flow obtains for all figures 
above the upper horizontal lines, the condition intermediate between 
weir and jet flow holds for all figures between the two sets of horizon¬ 
tal lines, while that of jet flow obtains for all figures below the second set 
of horizontal lines. 

At the point where the condition of weir flow changes to that of 
jet flow both of the above equations should theoretically hold true. 

















































210 


Chap. 7 . Flow of Water through Tubes 


By equating the second members of these equations the critical head 
at which the nature of the flow changes is found to be about 0.6 D for 
all values of 11 between 0.1 and 3.0 feet. Practically, however, the 
exact point at which the change occurs cannot be exactly determined. 

Prob. 87 . Compute the flow from a vertical pipe 14 inches in diameter 
when the head above the top of the pipe, as measured by a Pitot tube, is 
0.04 feet. Also compute the discharge when the head is 7.6 feet. 

Art. 88. Computations in Metric Measures 

Nearly all the formulas of this chapter are rational and may be 
used in all systems of measures. In the metric system lengths are to 
be taken in meters, areas in square meters, velocities in meters per 
second, discharges in cubic meters per second, and using for the accel¬ 
eration constants the values given in Table 9 c. 

(Art. 83 ) , The coefficients of discharge and velocity for smooth 
fire nozzles 2.0, 2.5, 3.0, and 3.5 centimeters in diameter are 0.983, 
0.972, 0.973, and 0.959, respectively. In using the formula ( 83) 2 
the values of d and /q should be taken in meters, but in finding the 
ratio D/d the values of D and d may be in centimeters or any other 
convenient unit. The constant g being 9.80 meters per second, the 
discharge q will be in cubic meters per second. When it is desired to 
use the gage reading p x in kilograms per square centimeter and to 
take D in centimeters, the formula 

'/ - 65.96 d D °\J i _ c / D / d y 

may be used for finding the discharge in liters per minute. 

Prob. 88a. Compute the loss of head which occurs when a pipe, dis¬ 
charging 18.5 cubic meters per second, suddenly enlarges in diameter from 
1.25 to 1.50 meters. 

Prob. 8 8 b. Find the coefficient of discharge for a tube 8 centimeters 
in diameter when the flow under a head of 4 meters is 18.37 cubic meters in 
5 minutes and 15 seconds. 

Prob. 88 c. Compute the discharge from a smooth nozzle 2.5 centimeters 
in diameter, attached to a hose 7.5 centimeters in diameter, when the pres¬ 
sure at the entrance is 5.2 kilograms per square centimeter. 




Fundamental Ideas. Art. 89 


211 


CHAPTER 8 

FLOW OF WATER THROUGH PIPES 
Art. 89 . Fundamental Ideas 

Pipes made of clay were used in very early times for convey¬ 
ing water. Pliny says that they were two digits (0.73 inches) in 
thickness, that the joints were filled with lime macerated in oil, 
and that a slope of at least one-fourth of an inch in a hundred 
feet was necessary in order to insure the free flow of water.* The 
Romans also used lead pipes for conveying water from their aque¬ 
ducts to small reservoirs and from the latter to their houses. 
Frontinus gives a list of twenty-five standard sizes of pipes,! 
varying in diameter from 0.9 to 9 inches, which were made by 
curving a sheet of lead about ten feet long and soldering the 
longitudinal joint. The Romans had confused ideas of the laws 
of flow in pipes, their method of water measurement being by 
the area of cross-section, with little attention to the head or pres¬ 
sure. They knew that the areas of circles varied as the squares 
of the diameters, and their unit of water measurement was the 
quinaria, this being a pipe i| digits in diameter; then the denaria 
pipe, which had a diameter of 2J digits, was supposed to deliver 
4 quinarias of water. 

In modern times lead pipes have also been used for house 
service, but these are now largely superseded by either iron pipes 
or iron pipes lined with lead or tin. For the mains of city water 
supplies cast-iron pipes are most common, and since 1890 steel- 
riveted pipes have come into use for large sizes. Lap-welded 
wrought-iron or steel pipes are used in some cases where the pres¬ 
sure is very high, and large wooden stave pipes are in use in the 
western part of the United States. 

* Natural History, book 31, chapter 31, line 5. 
f Herschel, Water Supply of the City of Rome (Boston, 1899), p. 36. 


212 


Chap. 8. Flow of Water through Pipes 


The simplest case of the flow of water through a pipe is that 
where the diameter of the pipe is constant and the discharge occurs 
entirely at the open end. This case will be discussed in Arts. 
90 - 99 , and afterwards will be considered the cases of pipes of 
varying diameter, a pipe with a nozzle at the end, and pipes with 
branches. Most of the principles governing the simple case 
apply with slight modification to the more complex ones. Pipes 
used in engineering practice range in diameter from ^ inch up to 
io feet or more. 

The phenomena of flow for this common case are apparently 
simple. The water from the reservoir, as it enters the pipe, meets 
with more or less resistance, depending upon the manner of con¬ 
necting, as in tubes (Art. 80 ). Resistances of friction and cohe- 



Fig. 89a. 



sion must then be overcome along the interior surface, so that the 
discharge at the end is much smaller than in the tube (Art. 84 ). 
When the flow becomes steady, the pipe is entirely filled through¬ 
out its length; and hence the mean velocity at any section is the 
same as that at the end, since the size is uniform. This velocity 
is found to decrease as the length of the pipe increases, other 
things being equal, and becomes very small for great lengths, 
which shows that nearly all the head has been lost in overcoming 
the resistances. The length of the pipe is measured along its 
axis, following all the curves, if there be any. The velocity con¬ 
sidered is the mean velocity, which is equal to the discharge di¬ 
vided by the area of the cross-section of the pipe. The actual 
velocities in the cross-section are greater than this mean near the 
center and less than it near the interior surface of the pipe, the 
law of distribution being that explained in Art. 86. 

The object of the discussion of flow in pipes is to enable the 
discharge which will occur under given conditions to be deter- 


































Fundamental Ideas. Art. 89 


213 


mined, or to ascertain the proper size which a pipe should have 
in order to deliver a given discharge. The subject cannot, how¬ 
ever, be developed with the definiteness which characterizes the 
flow from orifices and weirs, partly because the condition of the 
interior surface of the pipe greatly modifies the discharge, partly 
because of the lack of experimental data, and partly on account 
of defective theoretical knowledge regarding the laws of flow. 
In orifices and weirs errors of two or three percent may be re¬ 
garded as large with careful work; in pipes such errors are com¬ 
mon, and are generally exceeded in most practical investigations. 
It fortunately happens, however, that in most cases of the design 
of systems of pipes errors of five and ten percent are. not impor¬ 
tant, although they are of course to be avoided if possible, or, 
if not avoided, they should occur on the side of safety. 

The head which causes the flow is the difference in level from 
the surface of the water in the reservoir to the center of the end, 
when the discharge occurs freely into the air as in Fig. 89 a. If 
h be this head, and W the weight of water discharged per second, 
the theoretic potential energy per second is Wh; and if v be the 
actual mean velocity of discharge, the kinetic energy of the dis¬ 
charge is W • v 2 /2g. The difference between these is the energy 
which has been transformed into heat in overcoming the resist¬ 
ances. Thus the total head is /?, the velocity-head of the out¬ 
flowing stream is v 2 /2g , and the lost head is li — v 2 /2g. If the 
lower end of the pipe is submerged, as in Fig. 8 %, the head h is 
the difference in elevation between the two water levels. 

The total loss of head in a straight pipe of uniform size con¬ 
sists of two parts, as in a long tube (Art. 84 ). First, there is a 
loss of head h' due to entrance, which is the same as in a short 
cylindrical tube, and secondly there is a loss of head li" due to 
the frictional resistance of the interior surface. The loss of head 
at entrance is always less than the velocity-head and in this 
chapter it will be expressed by the formula 

h! = w— ( 89 ) i 

in which m is‘0.93 for the inward projecting pipe, 0.49 for the 


214 Chap. 8. Flow of Water through Pipes 

standard end, and o for a perfect mouthpiece, as shown in Art. 84 . 
When the condition of the end is not specified, the value used for 
m will be 0.5, which supposes that the arrangement is like the 
standard tube, or nearly so. For short pipes, however, it may 
be necessary to consider the particular condition of the end, and 
then m is to be computed from 

m = ( 1 /ci ) 2 — 1 (89) 2 

in which the coefficient c\ is to be selected from the evidence pre¬ 
sented in the last chapter. 

It should be noted that the loss of head at entrance is very 
small for long pipes. For example, it is proved by actual gagings 
that a clean cast-iron pipe 10 000 feet long and 1 foot in diameter 
discharges about 4J cubic feet per second under a head of 100 feet. 
The mean velocity then is, if q be the discharge and a the area of 
the cross-section, 

v = - = - - - -- = 5.41 feet per second, 
a 0.7854 

and the probable loss of head at entrance hence is 
h' = 0.5 X 0.01555 X 5.41 2 = 0.23 feet, 

or only one-fourth of one per cent of the total head. In this case 
the effective velocity-head of the issuing stream is only 0.45 feet, 
which shows that the total loss of head is 99.55 feet, of which 
99.32 feet are lost in friction. 

Prob. 89 . Under a head of 20 feet a pipe 1 inch in diameter and 100 
feet long discharges 15 gallons per minute. Compute the loss of head at 
entrance. 

Art. 90 . Loss of Head in Friction 

The loss of head due to the resisting friction of the interior 
surface of a pipe is usually large, and in long pipes it becomes very 
great, so that the discharge is only a small percentage of that due 
to the head. Let h be the total head on the end of the pipe where 
the discharge occurs, v 2 / 2 g the velocity-head of the issuing stream, 
h f the head lost at entrance, and h " the head lost in friction. Then 
if the pipe is straight, so that no other losses of head occur, 

h = ti + h" + — 

2 g 



Loss of Head in Friction. Art. 90 


215 


Inserting for the entrance-head ti its value from Art. 89 , this 

equation becomes _ l2 9 

h= m— + h" + — 

2 g 2 g 


which is a fundamental formula for the discussion of flow in 
straight pipes of uniform size. 

The head lost in friction may be determined for a particular 
case by measuring the head h, the area a of the cross-section of 
the pipe, and the discharge per second q. Then q divided by a 
gives the mean velocity v, and from the above equation, inserting 
for m its value from ( 89 ) 2 , there is found 

A"-A-V- 

Cl 2 g 

which serves to compute h ", the value of c\ being first selected 
according to the condition of the end. This method is not a good 
one for short pipes because of the uncertainty regarding the co¬ 
efficient Ci (Art. 84 ), but for long pipes it gives precise results. 


Another method, and the one most generally employed, is by 
the use of piezometers (Art. 85 ). A portion of the pipe being 
selected which is free from sharp curves, two piezometer tubes are 
inserted into which the water rises, or the pressure-heads are 
measured by gages (Art. 36 ). The difference of level of the water 
surfaces in the piezometer tubes is then the head lost in the pipe 
between them (Art. 85 ), and this loss is caused by friction alone 
if the pipe be straight and of uniform size. 

By these methods many observations have been made upon 
pipes of different sizes and lengths under different velocities of 
flow, and the discussion of these has enabled the approximate 
laws to be deduced which govern the loss of head in friction, and 
tables to be prepared for practical use. These laws are: 

1. The loss of head in friction is directly proportional to the 
length of the pipe. 

2. It is inversely proportional to the diameter of the pipe. 

3. It increases nearly as the square of the velocity. 

4. It is independent of the pressure of the water. 

5. It increases with the roughness of the interior surface. 


216 


Chap. 8. Flow of Water through Pipes 


These five laws may be expressed by the formula 


h 


// 


_ 

d 2 g 


(90) 



Fig. 90. 


in which / is the length of the pipe, d its diameter, / is an abstract 
number which depends upon the degree of roughness of the sur¬ 
face, and v 2 /2g is the velocity-head due to the mean velocity. 

This formula may be justified by reasonings based on the 
assumption that what has been called the loss in friction is really 
caused by impact of the particles of water against each other. 
Fig. 90 represents a pipe with the roughness of its surface enor¬ 
mously exaggerated and imperfectly 
shows the disturbances thereby 
caused. As any particle of water 
strikes a protuberance on the surface, 
it is deflected and its velocity dimin¬ 
ished, and then other particles of water in striking against it also 
undergo a diminution of velocity. Now in this case of impact the 
resisting force F acting over each square unit of the surface is to 
be regarded as varying with the square of the velocity (Arts. 27 
and 76 ). The total resisting friction for a pipe of length / and 
diameter d is then nrdlF, and the work lost in one second is dl'rrFv. 
Let W be the weight of water discharged in one second, then 
Wh" is also the energy lost in one second. But W = wq, \iw be 
the weight of a cubic unit of water and q the discharge per second, 
and the value of q is rdh. Then, equating the two expressions 
for the lost energy, and replacing F by Cv 2 where C is a constant, 
there results A j A r i 

h " = *t F = *±. v. 

w d w d 


Now C must increase with the roughness of the surface and hence 
this expression is the same in form as ( 90 ), and it agrees with the 
five laws of experience. 

Values of h" having been found by experiments, in the manner 
described above, values of the quantity / can be computed. In 
this way it has been found that / varies not only with the rough¬ 
ness of the interior surface of the pipe, but also with its diameter, 













































Loss of Head in Friction. Art. 90 


217 


and with the velocity of flow. From the discussions of Fanning. 
Smith, and others, the mean values of / given in Table 90<z have 
been compiled, which are applicable to clean cast-iron andwrought- 
iron pipes, either smooth or coated with coal-tar, and laid with 
close joints. 

Table 90a. Friction Factors for Clean Iron Pipes 


Diameter 

Velocity in Feet per Second 









Feet 









1 

2 

3 

4 

6 

io 

15 

0.05 

O.047 

0.041 

O.037 

O.034 

0.031 

0.029 

0.028 

O.I 

.038 

.032 

.030 

.028 

.026 

.024 

.023 

0.25 

.032 

.028 

.026 

.025 

.024 

.022 

.021 

o -5 

.028 

.026 

.025 

.023 

.022 

.020 

.OI9 

°-75 

.026 

.025 

.024 

.022 

.021 

.OI9 

.Ol8 

1. 

.025 

.024 

.023 

.022 

.020 

.Ol8 

.017 

1-25 

.024 

.023 

.022 

.021 

.019 

.017 

.Ol6 

i -5 

.023 

.022 

.021 

.020 

.Ol8 

.Ol6 

.015 

i -75 

.022 

.021 

.020 

.Ol8 

.017 

.015 

.014 

2 . 

.021 

.020 

.019 

.017 

.Ol6 

.014 

.013 

2-5 

.020 

.019 

.Ol8 

.Ol6 

.015 

.013 

.012 

3 - 

.019 

.Ol8 

.Ol6 

.015 

.014 

.OI3 

.012 

3-5 

.Ol8 

.017 

.Ol6 

.014 

.013 

.012 


4 - 

.017 

.Ol6 

.015 

.013 

.012 

.Oil 


5 - 

.Ol6 

.015 

.014 

.013 

.012 



6. 

.015 

.014 

.013 

.012 

.Oil 

s' 



The quantity/ may be called the friction factor, and the table 
shows that its value ranges from 0.05 to 0.01 for new clean iron 
pipes. A rough mean value, often used, is 

Friction factor /= 0.02 

It is seen that the tabular values of / decrease both when the 
diameter and when the velocity increases, and that they vary 
most rapidly for small pipes and low velocities. The probable 
error of a tabular value of / is about one unit in the third decimal 
place, which is equivalent to an uncertainty of 10 percent when 
/ = 0.011, and to 5 percent when / = 0.021. The effect of this 
is to render computed values of h" liable to the same uncertainties; 
but the effect upon computed velocities and discharges is much 
less, as will be seen in Art. 93 . 



















'218 


Chap. 8. Flow of Water through Pipes 


To determine, therefore, the probable loss of head in friction, 
the velocity v must be known, and / is taken from Table 90 a for 
the given diameter of pipes. The formula ( 90 ) then gives the 
probable loss of head in friction. For example, let l = io ooo 
feet, d = i foot, v = 5.41 feet per second. Then from Table 90 a 
the factor / is 0.021, and 

h = 0.021 X-X 0.455 = 95-5 feet, 

1 

which is to be regarded as an approximate value, liable to an 
uncertainty of 5 percent. 

Table 90 &. Friction Head for ioo Feet of Clean Iron Pipe 


Diameter 

in 

Feet 

Velocity in Feet per Second 

1 

2 

3 

4 

6 

10 

is 


Feet 

Feet 

Feet 

Feet 

Feet 

Feet 

Feet 

0.05 

1.46 

5.10 

10.3 

16.9 

34-7 



O.I 

0-59 

I.99 

4.20 

6.97 

14-5 

37-3 


O.25 

.20 

O.70 

1.46 

2.40 

5-37 

i 3-7 

29.4 

0-5 

.09 

•32 

0.70 

I.14 

2.46 

6.22 

13-3 

0-75 

•05 

.21 

•45 

o -73 

i -57 

3-94 

8.40 

I. 

.04 

•15 

•32 

•55 

1.12 

2.80 

5-95 

I -25 

•03 

.11 

.25 

.42 

0.85 

2.11 

4.48 

i-5 

.02 

.09 

.20 

•33 

.67 

1.66 

3-50 

i -75 

.02 

.07 

.l6 

.26 

•54 

i -33 

2.80 

2. 

.02 

.06 

A 3 

.21 

•45 

1.09 

2.27 

2-5 

.OI 

•05 

.IO 

.16 

•34 

0.81 

1.68 

3 - 

•OI 

.04 

.07 

.12 

.26 

.67 

1.40 

3-5 

.OI 

•03 

.06 

.10 

.21 

•53 


4 - 


.02 

•05 

.08 

•17 

.42 


5 . 


.02 

.04 

.06 

•13 



6. 


.OI 

•03 

•05 

.10 




From Table 90 a and formula ( 90 ) the losses of head in friction 
for 100 feet of clean cast-iron pipe have been computed for differ¬ 
ent values of d and / and are given in Table 90 />, from which ap¬ 
proximate computations may be rapidly made. Thus, for the 
above data, by interpolation in Table 906 , there is found 0.952 
feet for the loss in 100 feet of pipe, and then for 10 ooo feet the 
loss of head is 95.2 feet. 




















Loss of Head in Curvature. Art. 91 


219 


Prob. 90 . Determine the actual loss of head in friction from the fol¬ 
lowing experiment: l = 6o feet, h = 8.33 feet, d = 0.0878 feet, q = 0.03224 
cubic feet per second, and c = 0.8. Compute the probable loss for the same 
data from formula ( 90 ) and also from Table 90 b. 

Art. 91 . Loss of Head in Curvature 

Thus far the pipe has been regarded as straight, so that no 
losses of head occur except at entrance and in friction. But 
when the pipe is laid on a curve, the water suffers a change in 
direction whereby an increase of pressure is produced in the 
direction of the radius of the curve and away from its center 
(Art. 156 ). This increase in pressure causes eddying motions of 
the water, from which impact results and energy is transformed 
into heat. The total loss of head h!" due to any curve evidently 
increases with its length, and should be greater for a small pipe 
than for a large one. Hence the loss of head due to the curvature 
of a pipe may be written 

h'" =fi~— ( 91 ), 

d 2 g 

in which l is the length of the curve, d the diameter of the pipe, 
v the mean velocity of flow, and f x is an abstract number called 
the curve factor, that depends upon the ratio of the radius of the 
curve to the diameter of the pipe. Let R be the radius of the 
circle in which the center line of the pipe is laid. Then, if R is 
infinity, the pipe is straight and /1 = o; but as the ratio R/d 
decreases, the value of f x increases. 

There are few experiments from which to determine the values 
of J\. Weisbach, about 1850, from a discussion of his own ex¬ 
periments and those of Castel, deduced a formula for the value of 
Jd/d for curves of one-fourth of a circle* and from this the follow¬ 
ing values of the curve factor f x have been computed : 

for R f d= 20 10 5 3 2 1.5 i*o 

fi = 0.004 0.008 0.016 0.030 0.047 0.072 0.184 

These values of fi are applicable only to small smooth iron pipes 
where the entire curve is without joints, since most of the pipes 

* Die Experimentale Hydraulik (Freiberg, 1855), p. 159. Mechanics 
of Engineering (New \ork, 1870), vol. 1, p. 898. 



220 Chap. 8. Flow of Water through Pipes 

on which the above experiments were made were probably of 
this kind. 

Freeman, in 1889, made measurements of the loss of head in 
fire hose 2.49 and 2.64 inches in diameter, and the curves were 
complete circles of 2, 3, and 4 feet radius.* From the results 
given for the smaller hose the following values of the curve fac¬ 
tor fi have been found : 

iorR/d = 19.2 14.4 9.6 

fi = 0.0033 0.0034 0.0048 

while for the larger hose the values are 

iorR/d= 16.2 13.6 8.1 

fi = 0.0036 0.0046 0.0045 

These values are in fair agreement with those given above for the 
small iron pipes. 

Williams, Hubbell, and Fenkell, in 1898 and 1899, made meas¬ 
urements in Detroit on cast-iron water mains having curves of 
90°. From their results for a 30-inch pipe the values of the curve 
factor/1 have been computed and are found to be as follows: 

for R/d =24 16 10 6 4 2.4 

fi = 0.036 0.037 0.047 0.060 0.062 0.072 

while from their work on a 12-inch pipe the values are 

for R/d= 4 3 2 1 

fi = 0.05 0.06 0.06 0.20 

Of these values, those derived from the larger pipe are the most 
reliable, and it is seen that they are much greater than the values 
deduced from Weisbach’s investigations on small pipes. Prob¬ 
ably some of this increase is due to the circumstance that the 
curves had rougher surfaces and that the joints were nearer to¬ 
gether than on the straight portions. These experiments f were 
made with the Pitot tube in the manner explained in Arts. 41 and 
86. They show that the law of distribution of the velocities in 
the cross-section is quite different from that for a straight pipe, 

* Transactions American Society of Civil Engineers, 1889, vol. 21, p. 363. 
f Transactions American Society of Civil Engineers, 1902, vol. 47. 


Loss of Head in Curvature. Art. 91 


221 


the maximum velocity being not at the center, but between the 
center and the outside of the curve. 


From the experiments of Schoder,* on 6-inch pipe and bends 
of 90°, the following values of / have been computed for velocities 
of 5 and 16 feet per second: 

(or R/d = 20 15 10 6 5 2 

v = Sj fi = 0.008 0.004 0.010 0.020 0.018 0.049 

v=i 6 , J\= 0.008 0.009 o.oii 0.021 0.022 0.059 

The data given by Davis,* from his experiments on pipe about 
2 jq inches in diameter for bends of 90°, enable the following values 
of fi to be computed for velocities of 5 and 15 feet per second: 

for R/d = 10 6 5 4 2 1 

v= 5, fi = 0.023 0.024 0.027 0.032 0.081 0.323 

0=15, fi= 0.027 0.051 0.052 0.058 0.144 0.394 

From the experiments of Brightmore,f on pipes 4 inches in 
diameter and for bends of 90°, the values of fi given below have 
been computed for velocities of 5 and 10 feet per second: 

for R/d = 10 6 5 4 2 1 

5> /i = °- OI 3 °-°33 °-°34 0-036 °- io 5 °-4o6 

2=10, fi= 0.013 °-°34 0.040 0.046 0.127 0-365 


While the above values of fi are few in number, and not wholly 
in accord, yet they may serve as a basis for roughly estimating 
the loss of head due to curvature. For example, let there be 
two curves of 24 and 16 feet radius in a pipe 2 feet in diameter, 
each curve being a quadrant of a circle. The ratios R/d are 12 
and 8, and the values of f\, taken from those deduced above from 
the large Detroit pipe, are 0.044 and 0.053. The lengths of the 
curves are 37.7 and 25.1 feet, and then from (91 )i 


h nr = 0.044 


37-7 Z - 




O .83 


V~ 


h 


m 


0.053 




1 v 


V ? 


= 0.66 — 
2 g 2 g 


* Transactions American Society of Civil Engineers, vol. 52. 
f Proceedings Institution of Civil Engineers, vol. 169. 





222 Chap. 8. Flow of Water through Pipes 

are the losses of head for the two cases. Here it is seen that the 
easier curve gives the greater loss of head. By the use of the 
values of f\ deduced from Weisbach’s investigation, the loss of head 
is much smaller and the sharper curve gives the greater loss of 
head, since the coefficients of the velocity-head are found to be 
0.13 and 0.14 instead of 0.83 and 0.66. The subject of losses in 
curves is, indeed, in an uncertain state, since sufficient experiments 
have not been made either to definitely establish the validity of 
( 91 )i or to determine authoritative values of the curve factor f\. 
Probably it will be found that ]\ varies with the diameter d as well 
as with the ratio R/d. 

When there are several curves in a pipe line, the value of fi(l/d) 
for each curve is to be found and then these are to be added in 
order to find the total loss of head. Thus, in general, 

ti " = nh — (91)2 

n 

is the total loss of head, in which m\ represents the sum of the 
values of fi(l/d) for all the curves. It must be remembered, 
however, that this loss of head is occasioned by the fact that the 
pipe is curved and that it is to be added to the loss caused by 
friction along the entire length of the pipe. In other words the 
curve factor /1 does not include the friction factor /. 

The lost head due to curvature in a pipe line is usually low 
compared with that lost in friction, since the number of curves 
is usually made as small as possible. For example, take a pipe 
1000 feet long and 3 inches in diameter, which has ten curves, five 
being of 90° and 6 inches radius and five being of 57°.3 and 5 feet 
radius. From ( 90 ), using 0.02 for the mean friction factor, the 
loss of head in friction is 80 v 2 /2g. From ( 91 ) 1, using the curve 
factors deduced from Weisbach, the loss of head for the five sharp 
curves is 0.74 v 2 / 2 g, and that for the five easy curves is 0.4 v 2 /2g. 

Prob. 91 . If the central angle of a curve of 18 inches radius is 57°-3, 
what is the length of the curve? If a hose, 2\ inches in diameter, is laid 
on this curve, compute the loss in head due to curvature when the velocity 
in the hose is 30 feet per second and also when it is 15 feet per second. 



Other Losses of Head. Art. 92 


223 


Art. 92 . Other Losses of Head 


Thus far the cross-section of the pipe has been supposed to be 
constant, so that no losses of head occur except at entrance (Art. 
89 ), in friction (Art. 90 ), and in curvature (Art. 91 ). But if the 
pipe contains valves, or has obstructions in its cross-section, or is 
of different diameters, other losses occur which are now to be 
considered. 


The figures show three kinds of valves for regulating the flow 
in pipes: A being a valve consisting of a vertical sliding-gate, 
B a cock-valve formed by two rotating segments, and C a throttle- 
valve or circular disk which moves like a damper in a stovepipe. 




y-s/77s/77////////y///7jyz\ 



Y77/T7777Z77/77///7/7/777A Q 


The loss of head due to these may be very large when they are 

sufficiently closed so as to cause a sudden change in velocity. It 

may be expressed by ..... v 2 

h — m — 

2g 

in which m has the following values, as determined by Weisbach 
from his experiments on pipes of small diameter.* For the gate- 
valve let d' be the vertical distance that the gate is lowered below 
the top of the pipe; then 

C 71 ^ t A . 1 1 3 I A t 

lor a /a — o -g T 8 2 8 48 

m = 0.0 0.07 0.26 0.81 2.1 5.5 17 98 


For the cock-valve let 6 be the angle through which it is turned, 
as shown at B in Fig. 92 ; then 


0 

0 

<N 

O 

O 

1 —( 

0 

O 

II 

u 

£ 

3 °° 

0 

0 

50 ° 55 ° 

On 

O 

O 

6 5 ° 

m = o° 0.29 1.6 

5-5 

17 

53 106 

206 

486 

In like manner, for the throttle-valve the coefficients 

are : 


for 6= 5° io° 20 0 

3 °° 

40 ° 

S o° 6o° 

65° 

7 o° 

m — 0.24 0.52 1.5 

3-9 

11 

33 118 

256 

75 ° 

* Mechanics of Engineering, 

vol. I, 

Coxe’s 

translation, p. 

902. 





















224 


Chap. 8. Flow of Water through Pipes 


The number m hence rapidly increases and becomes very great 
when the valve is fully closed, but as the velocity is then zero 
there is no loss of head. The velocity v here, as in other cases, 
refers to that in the main part of the pipe, and not to that in the 
contracted section formed by the valve. 

Kuichling’s experiments * on a gate-valve for a 24-inch pipe 
give values of m which are somewhat greater than those deduced 
by Weisbach from pipes less than 2 inches in diameter. Con¬ 
sidering the great variation in size, the agreement is, however, 
a remarkable one. He found 


for rl' Id = i JL I 1 3 59 

ior a / a — 3 12 2 8 4 72 

m = 0.8 1.6 3.3 8.6 22.7 41.2 


and his computed value of m when d'/d equals f is 75.6. 

An accidental obstruction in a pipe may be regarded as causing 
a contraction of section, followed by a sudden expansion, and the 
loss of head due to it is, by Art. 76 , 



where a is the area of the section of the pipe, and a' that of the 
diminished section. This formula shows that when a' is one- 
half of a, the loss of head is equal to the velocity-head, and that 
m rapidly increases as a' diminishes. The same formula gives 
the loss of head due to the sudden enlargement of a pipe from the 
area a' to a. 

Air-valves are placed at high points on a pipe line in order to 
allow the escape of air that collects there. Mud-valves or blow- 
offs are placed at low points in order to clean out deposits that may 
be formed as well as to empty the pipe when necessary. These 
are arranged so as not to contract the section, and the losses of 
head caused by them are generally very small. When a blow- 
off pipe is opened and the water flows through it with the velocity 
v f the loss of head at its entrance, even when the edges are rounded, 
is as high as or higher than 0.56 v 2 /2g, according to the experi¬ 
ments of Fletcher. 


* Transactions American Society of Civil Engineers, 1892, vol. 26, p. 449. 




Formula for Mean Velocity. Art. 93 


225 


In the following pages the symbol h"" will be used to denote 

the sum of all the losses of head due to valves and contractions of 

section. Then , ,,2 

h = mt— (92) 

in which nu will denote the sum of all the values of m due to these 
causes. In case no mention is made regarding these sources of 
loss they are supposed not to exist, so that both m 2 and h”" are 
simply zero. 

Prob. 92 . Which causes the greater loss of head in a 24-inch pipe, 
a gate-valve one-half closed, or five 90° curves of 16 feet radius ? 


Art. 93. Formula for Mean Velocity 

The mean velocity in a pipe can now be deduced for the con¬ 
dition of steady flow. The total head being h, and the effective 
velocity-head of the issuing stream being v 2 /2g, the lost head is 
h — v 2 /2g, and this must be equal to the sum of its parts, or 


h 


zr 


2 g 


= ti + h" + ti" + h 


//// 


Substituting in this the values of the four lost heads, as de¬ 
termined in the four preceding articles, it becomes 


9 9 

, zr zr 

h -- = m — 


+ / + mi — + Wo 

2 g d 2g 2g 2g 


Zr 


2 £ 


and by solving for v there is found 


v 


2 gh 


1 1 1 (^3)i 

i ~r m J {L/d)+ nh + m 2 

which is the general formula for the mean velocity in a pipe of 
constant cross-section. 

The most common case is that of a pipe which has no curves, 
or curves of such large radius that their influence is very small, 
and which has no partially closed valves or other obstructions. 
For this case both ni\ and m 2 are zero, and, taking m as 0.5, the 

formula becomes _ 

2 gh 


v = J (93)* 

>/ r -5 +JQ/d) 

which applies to the great majority of cases in engineering practice. 











226 


Chap. 8. Flow of Water through Pipes 


In this formula the friction factor / is a function of v to be 
taken from Table 90 a, and hence v cannot be directly computed, 
but must be obtained by successive approximations. For exam¬ 
ple, let it be required to compute the velocity of discharge from 
a pipe 3000 feet long and 6 inches in diameter under a head of 9 
feet. Here l = 3000, d = 0.5, and h = 9 feet, and taking for 
/ the rough mean value 0.02, formula ( 93 ) 2 gives 

/ 2 X 32.l6 X 9 r , , 

v = \ —--= 2.2 feet per second. 

> 1.5 + 0.02 X 3000 X 2 

The approximate velocity is hence 2.2 feet per second and enter¬ 
ing the table with this, the value of / is found to be 0.026. Then 
the formula gives 


v = 


2 X 32.16 X 9 


= 1.92 feet per second. 


1.5 + 0.026 X 3000 X 2 

This is to be regarded as the probable value of the velocity, since 
the table gives / = 0.026 for v = 1.92. In this manner by one 
or two trials the value of v can be computed so as to agree with 
the corresponding value of /. 

To illustrate the use of the general formula ( 93 )i let the pipe 

in the above example be supposed to have forty 90° curves 

of 6 inches radius, and to contain two gate-valves which are half 

closed. Then from Arts. 91 and 92 there are found ni\ = 11.6 

for the curves and m* = 4.2 for the gates. The mean velocity 

then is / —--- 

_ / 2 X 32.16 X q 0 , 

v — \-T v — = 1.83 feet per second, 

>17.3 + 0.026X6000 ^ 

which is but a trifle less than that found before. With a shorter 
pipe, however, the influence of the curves and gates in retarding 
the flow would be more marked. 


The head required to produce a given velocity v can be ob¬ 
tained from ( 93 ) 1 or ( 93 ) 2 . Thus from the general formula the 
required head is 

h =(i + m +/(// d) + mi + nh) — 

in which for common computations m — 0.5, while ni\ and m2 are 
neglected. 









Computation of Discharge. Art. 94 


227 


The error in the computed velocity due to an error of one unit 
in the last decimal of the friction factor / is always relatively less than 
the error in / itself. For instance, where v is computed for the above 
example with / = 0.025, which is 4 percent less than 0.026, its 
value is found to be 1.96 feet per second, or 2 percent greater than 
1.92. In general the percentage of error in v is less than one-half 
of that in /. It hence appears that computed velocities are liable 
to probable errors ranging from 1 to 5 percent, owing to imperfections 
in the tabular values of / for new clean pipes. This uncertainty 
is as a rule still further increased by various causes, so that 5 per¬ 
cent is to be regarded as a common probable error in computations 
of velocity and discharge from pipes. 

Velocities greater than 15 feet per second are very unusual in 
pipes, and but little is known as to the values of / for such cases. For 
velocities less than 0.5 feet per second, the values of/ are also not known 
(Art. 110), so that only a rough reliance can be placed upon computa¬ 
tions. The usual velocity in water mains is less than five feet per 
second, it being found inadvisable to allow swifter flow on account 
of the great loss of head in friction. 

Prob. 93. Using for / the mean value 0.02 compute the head required 
to cause a velocity of 10 feet per second in a pipe 15 000 feet long and 18 
inches in diameter. 

Art. 94 . Computation of Discharge 

The discharge per second from a pipe of given diameter is 
found by multiplying the velocity of discharge by the area of the 
cross-section of the pipe, or 

q — rdh = 0.7854 drv ( 94 ) 

in which v is to be found by the method of the last article. 

For example, let it be required to find the discharge in gallons 
per minute from a clean pipe 3 inches in diameter and 1500 feet 
long under a head of 64 feet. Here d = 0.25, l = 1500, and h = 
64 feet. Then for / = 0.02 the velocity is found from ( 93) 2 to 
be 5.82 feet per second; then from Table 90 a is found / = 0.024 
and the velocity is 5.30 feet per second. The discharge in cubic 
feet per second is 

q = 0.7854 X 0.25 2 X 5.30 = 0.260 


228 


Chap. 8. Flow of Water through Pipes 


which is equal to 116.7 gallons per minute. This is the probable 
result, which is liable to the same uncertainty as the velocity, say 
about 3 percent; so that strictly the discharge should be written 
116.7 ± 3.6 gallons per minute. 

By inserting the value of v from ( 93 ) 2 in the above expression 
for q it becomes 

q = rd 2 

and from this the head required to produce a given discharge is 




TbS+fQ/d) 


r 

d A 


These formulas are not more convenient for precise computations 
than the separate expressions for v, q , and h previously estab¬ 
lished, since v must be computed in order to select/from the table. 
For approximate computations, however, when / may be taken 
as 0.02, they may advantageously be used. In the English system 
of measures li and d are to be taken in feet and q in cubic feet per 
second, and the constants in these two formulas have the values 


\'TT ~\/ 2g = 6.299 8/7T 2 g = 0.0252 

The last formula shows that the head required for a pipe of given 
diameter varies directly as the square of the proposed discharge. 
Thus, if a head of 50 feet delivers 8 cubic feet per second through 
a certain pipe, a head of about 200 feet will be necessary in order 
to obtain 16 cubic feet per second. 


Prob. 94 . What head is required to discharge 6 gallons per minute 
through a pipe 1 inch in diameter and 1000 feet long ? 


Art. 95 . Computation of Diameter 

It is an important practical problem to determine the diameter 
of a pipe to discharge a given quantity of water under a given 
head and length. The last equation above serves to solve this 
case, if the curve and valve resistances be omitted, as all the 
quantities in it except d are known. This equation reduces to 

< 2 5 = -f-(i-5<* + /04 

7 r z g h 






Computation of Diameter. Art. 95 


229 


and for the English system of measures this becomes 


d = 0.4789 


(i-s d+fl) 


h J 


( 95 ) 


which is the formula for computing d when h, /, and d are in feet 
and q is in cubic feet per second. The value of the friction factor 
/ may be taken as 0.02 in the first instance, and the d in the right- 
hand member being neglected, an approximate value of the diam¬ 
eter is computed. The velocity is next found by the formula 

v = q! a = q/o.y8$4.d 2 

and from the Table 90 u the value of / thereto corresponding is 
selected. The computation for d is then repeated, placing in the 
right-hand member the approximate value of d. Thus by one 
or two trials the diameter is computed which will very closely 
satisfy the given conditions. 

For example, let it be required to determine the diameter 
of a new pipe which will deliver 500 gallons per second, its length 
being 4500 feet and the head 24 feet. Here the discharge is 

q = 500/7.481 = 66.84 cubic feet per second. 

The approximate value of d then is 

. 02X450 0X 66.84^ _ feet . 

24 J 

From this the mean velocity of flow is 


d = 0.479 f 




v 


^•^4 -- =7.6 feet per second, 


0 -7854 X 3.35 s 

and from the table the value of / for this diameter and velocity 
is found to be 0.013. Then 


d = 0.479 


w v s66.8 4 2 T 

(1.5 x 3*35 + °- OI 3 X 45°o)-—— 

24 


from which d = 3.125 feet. With this value of d the velocity 
is now found to be 8.71 feet, so that no change results in the value 
of /. The required diameter of the pipe is therefore 3.1 feet, or 
about 37 inches; but as the regular market sizes of pipes furnish 
only 36 inches and 40 inches, one of these must be used, and it 
will be on the side of safety to select the larger. 









230 


Chap. 8. Flow of Water through Pipes 


It is very important, in determining the size of a pipe, to also 
consider that the interior surface may become rough by corrosion 
and incrustation, thus increasing the value of the friction factor 
and diminishing the discharge. It has been found that some 
waters deposit incrustations which in a few years render the values 
of / more than double those given in Table 90 a. In Art. 106 
will be found values of the friction factor as determined by ex¬ 
periment on various pipes of different ages. The increase in 
/ from these causes is not likely to be so great in a large pipe as in 
a small one, but it is not improbable that for the above example 
they might be sufficient to make / as large as 0.03. Applying this 
value to the computation of the diameter from the given data 
there is found d = 3.6 feet = about 43 inches. 

The sizes of iron pipes generally found in the market are J, J, 1, 
i\, if, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 27, 30, 36, 40, 44, and 48 inches, 
while intermediate and larger sizes must be made to order. The com¬ 
putation of the diameter is merely a guide to enable one of these sizes 
to be selected, and therefore it is entirely unnecessary that the numer¬ 
ical work should be carried to a high degree of precision. In fact, 
three-figure logarithms are usually sufficient to determine reliable 
values of d from formula ( 95 ). 

Prob. 95 . Compute the diameter of a pipe to deliver 50 gallons per 
minute under a head of 4 feet when its length is 500 feet. Also when its 
length is 5000 feet. 


Art. 96 . Short Pipes 

A pipe is said to be short when its length is less than about 
500 times its diameter, and very short when the length is less than 
about 50 diameters. In both cases the coefficient C\ should be 
estimated according to the condition of the upper end as precisely 
as possible, and the length / should not include the first three 
diameters of the pipe, as that portion properly belongs to the tube 
which is regarded as discharging into the pipe. In attempting 
to compute the discharge for such pipes, it is often found that the 
velocity is greater than given in Table 90 a, and hence that the 
friction factor /cannot be ascertained. For this reason no accu¬ 
rate estimate can be made of the discharge from short pipes under 


Long Pipes. Art. 97 


231 


high heads, and fortunately it is not often necessary to use them 
in engineering constructions. 

For example, let it be required to compute the velocity of 
flow from a pipe i foot in diameter and ioo feet long under a head 
of ioo feet, the upper end being so arranged that C\ = 0.80, and 
hence m = 0.56 (Art. 89 ). Neglecting m\ and m 2 , since the pipe 
has no curves or valves, formula ( 93 ) 1 for the velocity becomes 

1.56+/(//<*) 

and, using for / the rough mean value 0.02 and taking l as 97 feet, 
there is found 42.9 feet per second for the mean velocity. Now 
there is no experimental knowledge regarding the value of the 
friction factor / for such high velocities in iron pipes, but judging 
from the table it is probable that / may be about 0.015. Using 
this instead of 0.02 gives for v the value 46 feet per second. 

The general equation for the velocity of discharge deduced in 
Art. 93 may be applied to very short pipes by writing l — 3d in 
place of /, and placing for m its value in terms of the coefficient C\. 
It then becomes 

v = 


If in this l equals 3d, the velocity is C\ V2 gh, which is the same as 
for the short cylindrical tube. If l = 12 d,f = 0.02, and C\ = 0.82, 
it gives v = 0.774 V2 gh, which agrees well with the value given 
by Art. 84 for this case. If l = 60 d, it gives v = 0.613 ^ 2 sK 
which is 2 percent greater than the value given by Art. 84 . 

Prob. 96 . Compute the discharge per second for a pipe 1 inch in diam¬ 
eter and 40 inches long under a head of 4 feet. 

Art. 97 . Long Pipes 

For long pipes the loss of head at entrance becomes very small 
compared with that lost in friction, and the velocity-head is also 
small. Formula (93)2 for the mean velocity is 

2 S h _ 

1 . 5 +fd/d) 














232 


Chap. 8. Flow of Water through Pipes 


in which the first term in the denominator represents the effect 
of the velocity-head and the entrance-head, the mean value of 
the latter being 0.5. Now it may safely be assumed that 1.5 
may be neglected in comparison with the other term, when the 
error thus produced in v is less than 1 percent. Taking for j 
its mean value, this will be the case when 


Vi.s+0.02//^ -i l 

- ° ■ ■ — = 1.01, whence - = 3750 

V0.02 l/d d 

Therefore, when l is greater than about <\oood the pipe will be 
called long. 

For long pipes under uniform flow the velocity is found from 
the above equation by dropping 1.5, and the discharge is found 
by multiplying this mean velocity by the area of the cross-section. 
Hence the formulas for velocity and discharge are 


« =i7r J 2J jf (97)l 

which for the English system of measures becomes 


» = S.02 9 = 6.30 yjjj- ( 97 >, 

From these expressions for q the general and special formulas for 
computing the diameter of the pipe for a given discharge, length, 
and head are found to be 






These equations show that for very long pipes the discharge varies 
directly as the 2J power of the diameter, and inversely as the 
square root of the length. 

In the above formulas, d, h , and l are to be taken in feet, q 
in cubic feet per second, and / is to be found from Table 90 a, an 
approximate value of v being first obtained by taking / as 0.02. 
It should not be forgotten that computations of discharge or 
diameter from these formulas are liable to uncertainty on account 
of imperfect knowledge regarding the friction factors. Especially 
when the velocities are lower than one or higher than fifteen feet 














Long Pipes. Art. 97 


233 

per second the results obtained can be regarded as rough estimates 
only. The value of h in these formulas is really the friction-head 
h", since in their deduction the other heads, h!, h and h"", have 
been neglected as insensible. Hence when the diameter d , the 
length /, the total head //, and the discharge q have been measured 
for a long pipe the friction factor / may be computed. In this 
manner much of the data was obtained from which Table 90 <z 
has been compiled. 

For circular orifices and for short tubes of equal length under 
the same head, the discharge varies as the square of the diameter. 
For pipes of equal length under a given head the discharges vary 
more rapidly owing to the influence of friction, for formula ( 97 ) 2 

5 

shows that if / be constant, q varies as d 2 . The relative discharg¬ 
ing capacities of pipes hence vary approximately as the 2 J powers 
of their diameters. Thus, if two pipes of diameters d v and d 2 
have same length and head, and if qi and q 2 be their discharges, 

5 5 5 

qi/q 2 = di 1 /d 2 or q 2 = {d 2 /d\) 1 qi 

For example, if there be two pipes of 6 and 12 inches diameter, 
d 2 /di equals 2 and hence q 2 = 5.7*71, or the second pipe discharges 
nearly six times as much as the first. In a similar manner it can 
be shown that 32 pipes of 6 inches diameter have the same dis¬ 
charging capacity as 1 pipe 24 inches in diameter. 

When the variation in the friction factor is taken into account, 
the formula gives . 

?2 = giidi/diYi/i/fiY 

Now as the values of / vary not only with the diameter but with 
the velocity, a solution cannot be made except in particular cases. 
For the above example let the velocity be about 3 feet per second ; 
then from the table fi = 0.023 and f 2 = 0.019, and accordingly 

5 1 

q 2 = 2i(2) 2 (i.2) 2 = 6.2^1 

or the 12-inch pipe discharges more than six times as much as the 
6-inch pipe. 

Prob. 97 . Compute the diameter required to deliver 15 000 cubic feet 
per hour through a pipe 26 500 feet long under a head of 324.7 feet. It 
this quantity is carried in two pipes of equal diameter, what should be their 
size ? 


234 


Chap. 8. Flow of Water through Pipes 


Art. 98. Piezometer Measurements 

Let a piezometer tube be inserted into a pipe at any point D\ 
at the distance h from the reservoir measured along the pipe line. 
Let A\Di be the vertical depth of this point below the'water level 
of the reservoir; then if the flow be stopped at the end C, the 
water rises in the tube to the point A x . But when the flow occurs, 

the water level in the pie¬ 
zometer stands at some 
point Ci, and the pressure- 
head at Di is hi, or C\D\ 
in the figure. The distance 
AiCi then represents the 
velocity-head plus all the 
losses of head between Di 
and the reservoir. If no losses of head occur except at entrance 
and in friction, the value of A\C\ then is 

TJ V 2 . V 2 h V 2 

H i =- Ym -b / 7 — 

2 g 2 g d 2 g 

from which the piezometric height can be found when v has been 
determined by direct measurement or by gaging. 

For example, let the total length / = 3000 feet, d = 6 inches, 
h = 9 feet, and m = 0.5. Then, as in Art. 93, there is found 
/ = 0.026 and v = 1.917 feet per second. The position of the 
top of the piezometric column is then given by 

Hi = (1.5 + 0.052 li) X 0.05714 
and the height of that column above the pipe is 

hi = A1D1 - II, 

Thus if h = 1000 feet, Hi = 3.06 feet; and if / x = 2000 feet, 
Hi = 6.03 feet. If the pipe is so laid that A x Di is 9 feet, the cor¬ 
responding pressure-heads are then 5.94 and 2.97 feet. 

For a second piezometer inserted at D 2 at the distance / 2 from 
the entrance, the value II 2 is 

V 2 , V 2 , r l 2 V 2 



















Piezometer Measurements. Art. 98 


235 


Subtracting from this the expression for Hi, there is found 



( 98 ), 


The second member of this formula is the head lost in friction in 
the length l 2 — h (Art. 90 ), and the first member is the difference 
of the piezometer elevations. Thus is again proved the principle 
of Art. 85, that the difference of two piezometer elevations shows 
the head lost in the pipe between them ; in Art. 85 the elevations 
Hi and H 2 were measured upward from the datum plane, while 
here they have been measured downward from the water level 
in the reservoir. 

By the help of this principle the velocity of flow in a pipe may 
be approximately determined. A line of levels is run between 
the points D\ and D 2 , which are selected so that no sharp curves 
occur between them, and thus the difference H 2 — Hi is found, 
while the length l 2 — h is ascertained by careful chaining. Then, 
from the above formula, 



from which v can be computed by the help of the friction factors 
in Table 90 a. For example, Stearns, in 1880, made experiments 
on a conduit pipe 4 feet in diameter under different velocities of 
flow.* In experiment No. 2 the length l 2 — h was 1747.2 feet, 
and the difference of the piezometer levels was 1.243 f ee t- As¬ 
suming for / the mean value 0.02, and using 32.16 feet per second 
per second for g, the velocity was 



= 3.0 feet per second. 


This velocity in the table of friction factors gives / = 0.015 for 
a 4-foot pipe. Hence, repeating the computation, there is found 
v = 3.30 feet per second ; it is accordingly uncertain whether the 
value of / is 0.015 or 0.014. If the latter value be used, there is 
found v = 3.62 feet per second. The actual velocity, as deter¬ 
mined by measurement of the water over a weir, was 3.738 feet 


* Transactions American Society of Civil Engineers, 1885, vol. 14, p. 4 - 







236 Chap. 8. Flow of Water through Pipes 


per second, which shows that the computation is in error about 
4 percent. 

IV In order that accurate results may be obtained with piezom¬ 
eters it is necessary, particularly under low pressure-heads, that 
the tubes be inserted into the pipe at right angles. If they be 
inclined with or against the current, the pressure-head h will be 
greater or less than that due to the pressure at the mouth. Let 
be the angle between the direction of the flow and the inserted 
piezometer tube. Since the impulse in the direction of the cur¬ 
rent is proportional to the velocity-head (Art. 27 ), the component 
of this in the direction of the inserted tube tends to increase the 
normal pressure-height hi when 6 is less than 90° and to decrease 
it when 6 is greater than 90°. Thus 

ip" 

ho = hi + n — cos 6 
2 g 


may be written as approximately applicable to the two cases in 

which n is a coefficient 
the value of which has 
not been ascertained. 
In this, if the tube be 
inserted normal to the 
pipe, 6 = 90° and h 0 be¬ 
comes hi, the height 
due to the static pres¬ 
sure in the pipe; if v = o, the angle 6 has no effect upon the 
piezometer readings. But if 6 differs from 90° by a small angle, 
the error in the reading may be large when the velocity in the 
pipe is high. Fig. 98 /; illustrates the three cases. 



The question as to the point from which the pressure-head should 
be measured deserves consideration. In the fig¬ 
ures of preceding articles h x and \u have been esti¬ 
mated upward from the center of the pipe, and it 
is now to be shown that this is probably correct. 

Let Fig. 98 c represent a cross-section of a pipe to 
which are attached three piezometers as shown. If 
there be no velocity in the tube or pipe, the 



























































The Hydraulic Gradient. Art. 99 


237 


water surface stands at the same level in each piezometer, and 
the mean pressure-head is certainly the distance of that level above 
the center of the cross-section. If the water in the pipe be in motion, 
probably the same would hold true. Referring to formula ( 75 )i 
and to Fig. 75 a, it is also seen that if there be no velocity h' = hx~h 2 , 
which cannot be true unless h x — h 2 — o, since there can be no loss 
of head in the transmission of static pressures; hence hx and h 2 cannot 
be measured from the top of the section. In any event, since the pie¬ 
zometer heights represent the mean pressures, it appears that they 
should be reckoned upward from the center of the section. The pie¬ 
zometer couplings for hose devised by Freeman are arranged with con¬ 
nections on the top, bottom, and sides, as are also those used for the 
Venturi meter (Art. 38 ), and thus the results obtained correspond to 
mean pressures or pressure-heads. Even in cases where the two points 
of connection are so near together that the difference H2 — H 1, can be 
measured by a differential manometer (Art. 37 ), the method of con¬ 
necting the tubes to the pipes should receive careful attention. 

Prob. 98. At a point 500 feet from the reservoir, and 28 feet below its 
surface, a pressure gage reads 10.5 pounds per square inch; at a point 8500 
feet from the reservoir and 280.5 feet below its surface, it reads 61 pounds 
per square inch. If the pipe is 12 inches in diameter, compute the discharge. 

Art. 99. The Hydraulic Gradient 

The hydraulic gradient is a line which connects the water 
levels in piezometers placed at intervals along the pipe; or .rather, 
it is the line to which the 

water levels would rise if T_" 

yLJ J 

piezometer tubes were in- — - - r ~ 

serted. In Fig. 98 a the line 
BC is the hydraulic gradient, 
and it is now to be shown 
that for a pipe of uniform 
size this is approximately a 
straight line. For a pipe discharging freely into the air, as in Fig. 
98u, this line joins the outlet end with a point B near the top of 
the reservoir. For a pipe with submerged discharge, as in Fig. 
99u, it joins the lower water level with the point B. 

Let Di be any point on the pipe distant h from the reservoir, 



































238 


Chap. 8. Flow of Water through Pipes 


measured along the pipe line. The piezometer there placed 
rises to C h which is a point in the hydraulic gradient. The equa¬ 
tion of this line with reference to the origin A is given by the first 
equation of Art. 98 , or 

2 g d 2 g 

in which Hi is the ordinate AiCi, and h is the abscissa A A i, pro¬ 
vided that the length of the pipe is sensibly equivalent to its 
horizontal projection. In this equation the first term of the 
second member is constant for a given velocity, and is represented 
in the figure by AB or A\Bi ; the second term varies with l\, and 
is represented by B\C\. The gradient is therefore a straight line, 
subject to the provision that the pipe is laid approximately hori¬ 
zontal ; which is usually the case in practice, since quite material 
vertical variations may exist in long pipes without sensibly 
affecting the horizontal distances. 

When the variable point Di is taken at the outlet end of the 
pipe, Hi becomes the head //, and h becomes the total length /, 
agreeing with the formula of Art. 93 , if the losses of head due to 
curvature and valves be omitted. When di is taken very near 
the inlet end, li becomes zero and the ordinate Hi becomes AB, 
which represents the velocity-head plus the loss of head at en¬ 
trance to the pipe. 

When there are easy horizontal curves in a pipe line, the above 
conclusions are unaffected, except that the gradient BC is always 
vertically above the pipe, and therefore can be called straight 
only by courtesy, although as before the ordinate B x Ci is propor¬ 
tional to li. When there are sharp curves, the inclination of the 
hydraulic gradient becomes greater and it is depressed at each 
curve by an amount equal to the loss of head which there occurs. 
When an obstruction occurs in a pipe, or a valve is partially 
closed, there is a sudden depression of the gradient at the ob¬ 
struction or at the valve. 

If the pipe is so laid that a portion of it rises above the hy¬ 
draulic gradient as at A in Fig. 996 , an entire change of condition 
generally results. If the pipe is closed at C, all the piezometers 


The Hydraulic Gradient. Art. 99 


239 


stand in the line A A, at the same level as the surface of the reser¬ 
voir. When the valve at C is opened, the flow at first occurs 
under normal conditions, h being the head and BC the hydraulic 
gradient. The pressure-head 
at A is then negative, and 
represented by AA. As a 
consequence air tends to enter 
the pipe, and when it does so, 
owing to defective joints, the 
continuity of the flow is 
broken, and then the pipe from A to C is only partly filled 
with water. The hydraulic gradient is then shifted to BDi, the 
discharge occurs at A under the head Hi A, while the remain¬ 
der of the pipe acts merely as a channel to deliver the flow. It 
usually happens that this change results in a great diminution 
of the discharge, so that it has been necessary to dig up and relay 
portions of a pipe line which have been inadvertently run above 
the hydraulic gradient. This trouble can always be avoided by 
preparing a profile of the proposed route, drawing the hydraulic 
gradient upon it, and excavating the pipe trench well below the 
gradient. In cases where the cost of this excavation is so great 
that it is resolved to lay the pipe above the gradient, all the joints 
of the pipe above the gradient should be made absolutely tight 
so that no air can enter the pipe and interrupt the flow. 

When a large part of the pipe lies above the hydraulic gradient 
it is called a siphon. Conditions sometimes exist which require 
a pipe line to be laid as a siphon for a short distance. In such 
a case an air chamber is sometimes built at the highest elevation 
so that air may collect in it instead of in the pipe, and provision 
is made for recharging the siphon when the flow ceases by admit¬ 
ting water at the highest elevation, or by operating a suction- 
pump placed there, or by forcing water into the pipe by a pump 
located at a lower elevation. Probably the largest siphon ever 
constructed is that laid about 1885 at Kansas City, Mo., 
it being 42 inches in diameter, and 730 feet long, with the summit 
10 feet above the general level of the pipe line. The air that 










240 


Chap. 8. Flow of Water through Pipes 


collected at the summit was removed by operating a steam ejec¬ 
tor for a few minutes each day.* 


The pressure-head h\ at any point on the pipe line distant d 
from the reservoir may be expressed in terms of the static head on that 
point, the entrance-head //', and the friction-head h" by inspection 
of Fig. 99 a; thus, /„ = A Jh - h' - h" 


Further, from the similar triangles in the figure, 

h" = C h - h')(h/l) 

that is, the loss in friction in the distance l x is proportional to l v For 
long pipes, in which h' is small, this may be written h” = hfh /), 
or the friction loss at any point on the pipe line is proportional to the 
total head and to the distance of the point from the reservoir. 

The above discussion shows that it is immaterial where the pipe 
enters the reservoir, provided that it enters below the hydraulic 
gradient point B. It is also not to be forgotten that the whole inves¬ 
tigation rests on the assumption that the lengths l l and / are sensibly 
equal to their horizontal projections. 

Prob. 99 . A pipe 3 inches in diameter discharges 538 cubic feet per 
hour under a head of 12 feet. At a distance of 300 feet from the reservoir 
the depth of the pipe below the water surface in the reservoir is 4.5 feet. 
Compute the probable pressure-head at this point. 


Art. 100 . A Compound Pipe 


A compound pipe is one having different sizes in different 
portions of its length. The change from one length to another 
should be made by a “reducer,” which is a conical frustum several 
feet long, so that losses of head due to sudden enlargement or 
contraction are avoided (Arts. 76 , 77 ). Let d h d 2 , d 3 , etc., be the 
diameters; l h / 2 , / 3 , etc., the corresponding lengths, the total 
length being h + U + etc. Let v h v 2 , etc., be the velocities in 
the different sections. Neglecting the loss of head at entrance 
and also that lost in curvature, the total head h may be placed 
equal to the loss of head in friction, or 





-f etc. 


* Engineering News, 1891, vol. 26, p. 519; 1893, vol. 29, pp. 423, 588. 



A Compound Pipe. Art. 100 


241 


Now if the discharge per second be q, and the flow be steady 


= v/i ^i 2 % = q/\ Trd 2 2 , etc. 

Substituting these velocities and solving for q, gives 



( 100 ) 


in which the friction factors etc., corresponding to the given 
diameters and computed velocities are found from Table 90 a. 


For example, consider the 
case of a pipe having only 
two sizes; let d\ = 2 and 
1 \ = 2800 feet, d 2 = 1.5 and 
h = 2145 feet, and h = 127.5 
feet. Using for f x and f 2 the 
mean value, 0.02, and making 
there is found (/=26 . 2Cubic 



the substitutions in the formula, 
feet per second 


from which V\ = 8.3 and v 2 = 14.8 feet per second 

Now from Table 90 a it is seen that ]\ = 0.015 and f 2 — 0.015 5 and 

repeating the computation, 

q = 30.2 cubic feet per second 

whence v\ = 9.6 and v 2 = 17.1 feet per second. 

These results are probably as definite as the table of friction fac¬ 
tors will allow, but are to be regarded as liable to an uncertainty 
of several percent. 


To determine the diameter of a pipe which will give the same 
discharge as the compound one, it is only necessary to replace 
the denominator in the above value of q by fl/d°, where l — li + h 
+ etc., and d is the diameter required. Taking the values of 

/ as equal, this gives l , u 

— = —— -z~t etc 
d* d x b d 2 ' 


Applying this to the above example, it becomes 

4945 _ 2800 ■ 2145 
d b 2 5 1.5 5 

from which d = 1.68 feet, or about 20 inches. 























242 Chap. 8. Flow of Water through Pipes 

A compound pipe is sometimes used to prevent the hydraulic 
gradient from falling below the pipe line. Thus, it is seen in Fig. 
100 that the hydraulic gradient rises at and falls at D 2 , and that its 
slope over the larger pipe is less than over the smaller one. These 
slopes and the amount of rise at D 1 can be computed for a given 
case. Using the above numerical data, the loss of head in friction for 
ioo feet of the large pipe is 

7 // IOO V\ r . 

h =0.015-~ = 1.07 feet, 

2 2 g 

while the same for the small pipe is 4.55 feet. Hence the slope of the 
gradients AC X and C 2 C is more than four times as rapid as that of the 
gradient E X E 2 . In the large pipe at D x the velocity-head is 0.01555 
X 9-6 2 = 1.43 feet, and, supposing that no loss occurs in the reducer, 
the velocity-head for the small pipe is 4.55 feet. The vertical rise 
C X E X of the hydraulic gradient at D l is hence the rise in pressure-head 
4.55—1.43 = 3.12 feet, and a fall of equal amount occurs at D 2 . 

When a portion of a small pipe is to be replaced by a large one, it 
is immaterial in what part of the length it is introduced, for it is seen 
that formula (100) takes no note of where the length l A is placed in the 
total distance /. The Romans knew that an increase in the diameter 
of a pipe after leaving the reservoir would increase the discharge, and 
the law passed by the Roman senate about the year 10 b.c. forbade a 
consumer to attach a larger pipe to the standard pipe within 50 feet 
of the reservoir to which the latter was connected.* 

Prob. 100 . At Rochester, N.Y., there is a pipe 102 277 feet long, of 
which 50 828 feet is 36 inches in diameter and 51 449 feet is 24 inches in di¬ 
ameter. Under a head of 143.8 feet this pipe is said to have discharged in 
1876 about 14 cubic feet per second and in 1890 about io| cubic feet per 
second. Compute the discharge by ( 100 ), and draw the hydraulic gradient. 

Art. 101. A Pipe with a Nozzle 

Water is often delivered through a nozzle in order to perform 
work upon a motor or for the purposes of hydraulic mining, the 
nozzle being attached to the end of a pipe which brings the flow 
from a reservoir. In such a case it is desirable that the pressure 
at the entrance to the nozzle should be as great as possible, and 


* Herschel, Water Supply of the City of Rome (Boston, 1889), p. 77. 


A Pipe with a Nozzle. Art. 101 


243 


this will be effected when the loss of head in the pipe is as small 
as possible. Ihe pressure column in a piezometer, supposed to 


be inserted at the end of the 
pipe, as shown at CiA in 
Fig. 101 , measures the pres¬ 
sure-head there acting, and the 
height A\C\ measures the lost 
head plus the velocity-head, 
the latter being very small. 


- di ___ A 



Fig. 101 . 


Let h be the total head on the end of the nozzle, D its diameter, 
and V the velocity of the issuing stream. Let d and v be the 
corresponding quantities for the pipe, and l its length. Then the 
effective velocity-head of the issuing stream is V 2 / 2g, and the lost 
head is h — V' 2 / 2g. This lost head consists of several parts : 
that lost at the entrance D ; that lost in friction in the pipe; that 
lost in curves and valves, if any; and lastly, that lost in the nozzle. 
Then the principle of energy gives the equation 


V 2 v 2 


/ " 1 /1 v 2 , v 2 , v 2 1 / V 2 

n -- m - h/-—■ + m,i -r nu -b m 

2g 2 g d 2g 2 g 2g 2g 


Here m is determined by Art. 89 , / by Art. 90 , m\ by Art. 91 , m 2 
by Art. 92 , while m' for the nozzle is found in the same manner 
as m is found for the pipe, or m' = (1/ci) 2 — 1, where c\ is the co¬ 
efficient of velocity for the nozzle (Art. 83 ). This value of m' 
takes account of all losses of head in the nozzle, so that it is un¬ 
necessary to consider its length; for a perfect nozzle C\ is unity 
and m' is zero. 


The velocities v and V are inversely as the areas of the cor¬ 
responding cross-sections (Art. 31 ), since the flow is steady, 
whence V = v(d/D) 2 . Inserting this in the above equation and 
solving for v gives, if m\ and be neglected, 


v = 



_ 2 gh _ 

+/(//rf) + (i/c 1 )W £) 4 


( 101 ) 


for the velocity in the pipe. The velocity and discharge from the 
nozzle are then given by 

V = (d/DYv 


q = j 7 tD 2 V 
























244 


Chap. 8. Flow of Water through Pipes 


and the velocity head of the jet is V 2 /2g. These equations 
show that the greatest value of V obtains when D is as small as 
possible compared to d, and that the greatest discharge occurs 
when D is equal to d. When the object of a nozzle is to utilize 
the velocity-head of a jet, a large pipe and a small nozzle should 
be employed. When the object is to utilize the energy of the jet 
in producing power by a water wheel, there is a certain relation 
between D and d that renders this a maximum (Art. 161 ). 


As a numerical example, the effect of attaching a nozzle 
to the pipe whose discharge was computed in Art. 94 will be 
considered. There /= 1500, d = 0.25, and // = 64 feet; m — 0.5, 
v = 5.3 feet, and <7 = 0.26 cubic feet per second. Now let the nozzle 
be one inch in diameter at the small end, or D — 0.0833 f ee b and 
let its coefficient c\ be 0.98. Here d/D = $. and for /=0.025 
the velocity in the pipe is 


v = 


2 X 32.16 X 64 


0.5 + 0.025 X 1500 X 4 -f 1.041 X 81 


or v = 4.2 feet per second. The effect of the nozzle, therefore, 
is to reduce the velocity in the pipe. The velocity of the jet 
at the end of the nozzle is, however, 

V = v(d/D ) 2 = 37.8 feet per second, 
and the discharge per second from the nozzle is 

q = \ ttD 2 V = 0.206 cubic feet 

which is about 20 percent less than that of the pipe before the 
nozzle was attached. The nozzle, however, produces a marvel¬ 
ous effect in increasing the energy of the discharge ; for the veloc¬ 
ity-head corresponding to 5.3 feet per second is only 0.44 feet, 
while that corresponding to 37.8 feet per second is 22.2 feet, or 
about 50 times as great. As the total head is 64 feet, the efficiency 
of the pipe and nozzle is about 35 percent. 

If the pressure-head hi at the entrance of the nozzle be observed, 
either by a piezometer tube or by a pressure gage, the velocity of dis¬ 
charge from the nozzle can be computed by the formula 

v= I _£1*1_ 

V(i /c,y-(D/dy 







House-service Pipes. Art. 102 


245 


the demonstration of which is given in Art. 83 . This can be used when 
a hose and nozzle is attached at any point of a pipe or at a hydrant. 
It can also be used to compute h x when V has been found. Thus, for 
the above example, 



d*J 2g 


22.8 feet 


which shows that the loss of head in the nozzle is about o.6 feet. The 
loss of head at entrance, for this case, is about 0.2 feet, and the loss of 
head in friction in the pipe is 41.0 feet. 

Prob. 101 . A pipe 12 inches in diameter and 4320 feet long leads from 
a reservoir to a gravel bank against which water is delivered from a nozzle 
2 inches in diameter. The head on the end of the nozzle is 320 feet and the 
coefficient of velocity of the nozzle is 0.97. Compute the velocity in the 
pipe, the velocity-head of the jet, and the discharge. 


Art. 102. House-service Pipes 


A service pipe which runs from a street main to a house is 
connected to the former at right angles, and usually by a corpo¬ 


ration cock or by a “ ferrule. ’ 
in such cases is hence larger 
than in those before discussed, 
and m should probably be taken 
as at least equal to unity. The 
pipe, if of lead, is frequently 
carried around sharp corners by 
curves of small radius; if of 
iron, these curves are formed by 
pieces forming a quadrant of a 


The loss of head at entrance 



circle into which the straight parts are screwed, the radius of 
the center line of the curve being but little larger than the radius 
of the pipe, so that each curve causes a loss of head equal 
nearly to double the velocity-head (Art. 91 ). For new iron 
pipes the loss of head due to friction may be estimated by the 
rules of Art. 90 or by Table 906 . 

A water main should be so designed that a certain minimum 
pressure-head hi exists in it at times of heaviest draft. This 
pressure-head may be represented by the height of the pie- 















246 


Chap. 8. Flow of Water through Pipes 


zometer column AB , which would rise in a tube supposed to be 
inserted in the main, as in Fig. 102 u. The head h which causes 
the flow in the pipe is then the difference in level between the top 
of this column and the end of the pipe, or AC. Inserting for 
h this value, the formulas of Arts. 94 and 95 may be applied to the 
investigation of service pipes in the manner there illustrated. 
As the sizes of common house-service pipes are regulated by 
the practice of the plumbers and by the market sizes obtain¬ 
able, it is not often necessary to make computations regarding 
the flow of water through them. 

The velocity of flow in the main has no direct influence 
upon that in the pipe, since the connection is made at right angles. 
But as that velocity varies, owing to the varying draft upon the 
main, the effective head h is subject to continual fluctuations. 
When there is no flow in the main, the piezometer column rises 
until its top is on the same level as the surface of the reservoir; 
in times of great draft it may sink below C, so that no water can 
be drawn from the service pipe. 


The detection and prevention of the waste of water by con¬ 
sumers is a matter of importance in cities where the supply is 
limited and where meters are not in use. Of the many methods 
devised to detect this waste, one by the use of piezometers may 
be noticed, by which an inspector without entering a house may 
ascertain whether water is being drawn within, and the approxi¬ 


mate amount per second. Let M be the street main from which 
a service pipe MOH runs to a house II. At the edge of the side¬ 
walk a tube OP is connected to the service pipe, which has a three- 

way cock at O, which can be turned from 
above. The inspector, passing on his rounds 
in the night-time, attaches a pressure gage 
at P and turns the cock O so as to shut off 
the water from the house and allow the full 
pressure of the main p x to be registered. 
Then he turns the cock so that the water may flow into the 
house, while it also rises in OP and registers the pressure p 2 . 
Then if p 2 is less than pi, it is certain that waste is occurring 









House-service Pipes. Art. 102 


247 


within the house, and the amount of this may be approximately 
computed and the consumer be notified accordingly. 

The pitometer, which consists of a rated Pitot tube (Art. 
41 ), facing the current in the pipe, with a differential gage (Art. 
37 ) to determine the pressure-head due to the current, is also used 
for the measurement of the flow in water mains and for the detec¬ 
tion of water waste. A photographic record of the difference in 
height of the columns of liquid in the gage tube is kept, and this 
shows the discharge through the water main at any instant, 
as also all fluctuations in the flow.* (See Art. 38 .) 


w 


_ 


~L 


C 


B 


Fig. 102 c. 


When the pressure in the street main is very high, a pressure 
regulator may be placed between the main and the house in order 
to reduce the pressure and thus allow lighter pipes to be used in 
the house. Fig. 102 c shows the principle of its action, where 
A represents the pipe from the main 
and B the pipe leading to the house. 

A weight W is placed upon a piston 
which covers the opening into the 
chamber C. This weight and that of 

. A 

the piston are sufficient to overcome a --- 

certain unit-pressure in C, and therefore 
the unit-pressure in B is less than that 

in A by that amount. For example, suppose the pressure in A 
to be ioo pounds per square inch, and let it be required that 
the pressure in B shall not rise above 6o pounds per square 
inch; then the piston must be so weighted that it may exert on 
the water in C a pressure of 40 pounds per square inch. When 
water is drawn out anywhere along the pipe B, the pressure in 
the chamber above the piston falls below 60 pounds per square 
inch, and hence the piston rises and water flows from A into B 
until the pressure is restored. Instead of a weight, a spring is 
generally used, or sometimes a weighted lever. 

Large-sized pressure regulators are also used to control and 
maintain a constant pressure in distributing mains in cases where 


* Engineering Record, 1903, vol. 47, p. 122. 


















248 


Chap. 8. Flow of Water through Pipes 


a low service level is fed from one of higher pressure, or in situa¬ 
tions where it is desired to maintain a pressure which shall not 
exceed a fixed maximum. 

Prob. 102 . In Fig. 1026 let the house pipe be one inch in diameter and 
the pressure at the gage be 34 pounds per square inch when there is no flow. 
The distance from the-main to the gage is 16 feet and from the gage to the 
end of the pipe is 29 feet. At the end of the pipe, which is 5 feet higher than 
the gage, 2.1 gallons of water are drawn per minute. Compute the pressure 
at the gage. 


Art. 103 . Operating and Regulating Devices 

In the operation of nearly every water works system certain 
special apparatus is employed in order to maintain nearly con¬ 
stant conditions within the system and under the variable 
draft to which it is subjected. These forms of apparatus are 
designed to operate automatically and so to do away with hand 
regulation. Many of these are designed, as described under 
meters in Art. 38 , to trace on a chart a continuous autographic 
record of the pressure, of the water level, or of the discharge. 
Among these are pressure gages (Art. 36 ), water stage registers 
(Art. 34 ), and rate of flow' gages (Art. 38 ). 

Air valves are attached to water mains in situations where 
air is likely to accumulate within the pipe and by its presence in¬ 
terfere with the flow of the water or be carried along within the 
pipe and produce dangerous water hammer. Valves of this type 
permit the air within the pipe to escape, but automatically close 
and prevent the passage of water. They are also placed on all 
of the principal summits of riveted steel and other pipes so as to 
admit air into the pipe in case of a sudden break and thus pre¬ 
vent its collapse under external atmospheric pressure. In the 
case of cast-iron pipes, on account of the strength of their shells, 
this precaution is not usually necessary. The principle of the 
operation of the air valve is simply that of a float placed in a cham¬ 
ber above and connected with the pipe from which the air is to be 
removed. When air accumulates in the pipe, it passes up into 
the chamber; the float falls, and in falling, by means of a lever, 
operates and opens a valve. The air then escapes under the 


Operating and Regulating Devices. Art. 103 249 

pressure of the water until the float again rises and causes the 
valve to close. 

Pressure regulators operating on the principle described in 
Art. 102 are employed for the purpose of controlling and maintain¬ 
ing a constant pressure in distributing systems in situations where 
a low service level is fed from one of higher pressure. They 
may also be used to regulate the flow between reservoirs situated 
at different elevations. In the larger sized regulators the valve 
which controls the flow is operated by a pair of differential pistons 
connecting with a chamber, the pressure in which is caused to 
vary with fluctuations in pressure on the two sides of the regu¬ 
lator. The variations in pressure within this chamber are in¬ 
tensified by two small-sized regulators which connect directly 
to the high and low pressure sides of the large regulator. That 
on the upstream side of the main regulator is designed to close 
under an increase in pressure, while that on the downstream side 
will tend to open as the pressure rises. The effect of any dif¬ 
ference in pressure on the two sides of the main regulator is there¬ 
fore promptly reflected in the pressure within the chamber, and 
the differential pistons at once move to open or close the regu¬ 
lating valve in the effort to maintain within the pipe the pre¬ 
determined constant pressure at which the apparatus has been 
set. A sixteen-inch regulator of this type will control the pres¬ 
sure within narrow limits and pass through it, as may be necessary 
to accomplish this purpose, quantities up to io or 15 millions of 
gallons per day. 

Relief valves for the purpose of preventing the pressure within 
a pipe from rising above some predetermined limit, either on ac¬ 
count of a sudden falling off of the draft or by water hammer, are 
also made to operate on the principle described in Art. 102 , but 
in the reverse direction. The regulating valve described in 
the preceding paragraph may also be adapted for this use by 
simply making the necessary adjustments of the small regulators. 

In certain situations and principally in connection with the oper¬ 
ation of filtration plants it is desirable that the flow within a pipe 
shall be maintained at a constant rate. This may be accomplished 


250 Chap. 8. Flow of Water through Pipes 

by permitting the water to pass into an open chamber, from which it 
flows over and through a circular weir supported on floats. As the 
water rises in the chamber the weir also rises, and a constant relation 
is thus obtained between the height of the water and that of the weir 
crest. In order to limit the necessary height of the chamber the float 
maybe made to operate a butterfly valve on the inlet pipe, so that when 
the float rises the valve will partly close and thus diminish the quantity 
of water entering the chamber. Conversely as the float falls the 
valve is opened and more water permitted to enter. In neither of 
these two cases can the flow in the outlet pipe exceed the predeter¬ 
mined capacity of the circular weir. Another form of the rate of flow 
controller is that in which a balanced valve is operated by the differ¬ 
ences in pressure at the throat and downstream end of a Venturi 
tube inserted in the line. This valve will open or close as the quantity 
of water decreases or increases below or above some fixed quantity. 
In this manner a smaller or greater loss of head is automatically 
introduced into the system, and since the discharge is proportional 
to the square root of the effective head, the mechanism operates in 
such a manner as to maintain a constant flow. 

For determining the discharge or rate of flow within a pipe at 
any instant either a Venturi meter or a Pitot tube with the neces¬ 
sary connections may be used, as described in Arts. 38 and 41 . 

Loss of head gages are used in cases where it is desired to indicate 
at one place the loss of head which occurs between two points on a 
system. The most usual application is in the case of a filter bed 
where the loss of head is constantly varying on account of the clogging 
of the filter surface. In this situation a loss of head gage indicates 
at once whether or not a filter should be put out of service and cleaned. 
A gage for this service consists of a float in each of two chambers, 
the chambers being connected with the pipe or filter system at the 
points between which it is desired to measure the difference or loss of 
head. One of the floats is connected by means of a wire to a hori¬ 
zontal axis which carries a pointer, while the other is connected to 
another horizontal axis which carries the dial on which the pointer 
indicates. The two horizontal axes are concident, and the reading of 
the pointer indicates the loss of head. If the water in both of the cham¬ 
bers rises or falls an equal amount, the pointer will still indicate the 
same loss of head, as the directions of rotation of the pointer and dial 
are the same. In order to avoid a movable dial other forms of this 


Water Mains in Towns. Art. 104 


251 


apparatus are arranged by the introduction of a differential mechanism, 
so that the loss of head is directly indicated by the pointer on a sta¬ 
tionary dial. 

Valves for maintaining a constant level in a tank or reservoir are 
usually constructed, for small sizes, of a ball float operating a cock 
as it rises and falls by means of a system of levers. On larger work 
an ordinary gate valve operated by a hydraulic cylinder and piston 
may be used. A float either on the water surface itself or on the sur¬ 
face of mercury in a vessel connecting with the water operates a small 
three-way valve which admits the water either above or below the pis¬ 
ton of the hydraulic valve and so either closes or opens it as the water 
level rises above or falls below a fixed elevation. In order to prevent 
such valves from closing too rapidly and thus inducing water hammer, 
the ports of the three-way valve may be made quite small so as to 
cause the water to pass very slowly into the operating cylinder or else 
another piston may be introduced into the system and so arranged 
that the water behind it is permitted to escape through an orifice the 
size of which can be regulated. By this means the time of closing can 
be very nicely adjusted. 

All automatic devices are more or less likely to get out of order. 
This is simply due to the inherent difficulty in attaining perfection 
in any device. In order that they may at all times retain their ad¬ 
justment and properly perform the functions for which they have been 
designed they must be frequently inspected and always kept in good 
condition and repair. The selection of any particular form of regu¬ 
lating, control, or recording device will depend upon the conditions 
under which it is to operate and upon the past performance of the 
mechanism as attested by the experience of those who have used it. 

Prob. 103 . Make a sketch showing the arrangement above described 
for maintaining a constant level in a tank by means of a gate valve operated 
by a hydraulic cylinder. Show also the arrangement of the dampening 
piston for preventing too rapid closing of the valve. 

Art. 104. Water Mains in Towns 

The simplest case of the distribution of water is that where 
a single main is tapped by a number of service pipes near its end, 
as shown in Fig. 104 . In designing such a main the principal 
consideration is that it should be large enough so that the pres- 


252 


Chap. 8. Flow of Water through Pipes 


sure-head hi, when all the pipes are in draft, shall be amply suffi¬ 
cient to deliver the water into the highest houses along the line. 

It is generally recommended that 
this pressure-head in commercial and 
manufacturing districts should not 
be less than 150 feet, and in sub¬ 
urban districts not less than 100 feet. 
The height H to the surface of the 
water in the reservoir will always be 
greater than hi, and the pipe is to 
be so designed that the losses of head may not reduce hi below 
the limit assigned. The head h to be used in the formulas is 
the difference H — hi. The discharge per second q being known 
or assumed, the problem is to determine the proper diameter d 
of the water main. 


H 


hi 


0*0 o o 1 


.-*T— 


-r> 


Fig. 104 . 


A strict theoretical solution of even this simple case leads to 
very complicated calculations, and in fact cannot be made with¬ 
out knowing all the circumstances regarding each of the service 
pipes. Considering that the result of the computation is merely 
to enable one of the market sizes to be selected, it is plain that 
great precision cannot be expected, and that approximate methods 
may be used to give a solution entirely satisfactory. It will then 
be assumed that the service pipes are connected with the main 
at equal intervals, and that the discharge through each is the same 
under maximum draft. The velocity v in the main then decreases 
and becomes o at the dead end. The loss of head per linear foot 
in the length li (Fig. 104 ) is hence less than in l. To determine 
the total loss of head in the length l h let i\ be the velocity at a 
distance x from the dead end; then = v • x/li and the loss of 
head in friction in the length Sx is 


9 9 

X 1 V~ 


Bh"=f—^= f 

d 2 g ' dh 2 2 g 


Sx 


and hence between the limits o and h that loss of head is 


h"=fkt. 
' id 2 g 


(104) 





























Water Mains in Towns. Art. 104 


253 


provided that/ remains constant. This is really not the case, but 
no material error is thus introduced, since / must be taken larger 
than the tabular values in order to allow for the deterioration of 
the inner surface of the main. The loss of head in friction for 
a pipe which discharges uniformly along its length may therefore 
be taken at one-third of that which occurs when the discharge 
is entirely at the end. 


Now neglecting the loss of head at entrance and the effective 
velocity-head of the discharge, the total head h is entirely con¬ 
sumed in friction, or 



h if, 
3 d 2 g 


Placing in this for v its value in terms of the total discharge q 
and the diameter of the pipe, and solving for d, gives 


<F = (l + il i) 


i 6 /< 7 2 

2 g 7 T 2 h 


This is the same as the formula of Art. 97 , except that l has been 
replace by l + J/i. The diameter in feet then is 

[d- 04790 ' 


when h and l are in feet and q in cubic feet per second. 


For example, consider a village consisting of a single street 
with length li = 3000 feet, and upon which there are 100 houses, 
each furnished with a service pipe. The probable population 
is then 500, and taking 100 gallons per day as the consumption 
per capita, this gives for the average discharge per second along 
the length li 


Q = 


500 X 100 


7.48 X 3600 X 24 


= 0.0774 cubic feet, 


and since the maximum draft is often double of the average, 
q will be taken as 0.15 cubic feet per second. The length l to 
the reservoir is 4290 feet, w r hose surface is 90.5 feet above the dead 
end of the main, and it is required that under full draft the pres¬ 
sure-head in the main shall be 75 feet. Then h = 90.5 — 75 = 




, 254 


Chap. 8. Flow of Water through Pipes 


15.5 feet, and taking/ = 0.03 in order to be on the safe side, the 
formula gives ^ _ 0 ^6 feet = ^ ^ inches. 

Accordingly a four-inch pipe is nearly large enough to satisfy 
the imposed conditions. 

To consider the effect of lire service upon the diameter of 
the main, let there be four hydrants placed at equal intervals 
along the line /1, each of which is required to deliver 20 cubic 
feet per minute under the same pressure-head of 75 feet. This 
gives a discharge 1.33 cubic feet per second, or, in total, q = 1.33 
+ 0.15 = 1.5 cubic feet. Inserting this in the formula, and 
using for / the same value as before, 

d = 0.897 feet = 10.8 inches. 

Hence a ten-inch pipe is at least required to maintain the required 
pressure when the four hydrants are in full draft at the same time 
with the service pipes. 

Prob. 104 . Compute the velocity v and the pressure-head hi for the 
above example, if the main is 8 inches in diameter and the discharge be 1.5 
cubic feet per second. Also when the main is 12 inches in diameter. 


Art. 105 . Branches and Diversions 


In Fig. 105 a is shown a main of length l and diameter d , con¬ 
nected with a storage reservoir, which has two branches with 

lengths h and l 2 , and 
diameters d 1 and d 2 
leading to two smaller 
distributing reservoirs. 
These data being given, 
as also the heads H 1 
and II 2 under which 
the flow occurs, it is required to find the discharges q Y and q 2 . 
Let v, Vi, and v 2 be the corresponding velocities; then for long 
pipes, in which all losses except those due to friction may be 
neglected, the friction-heads for the two branches are 

h Vi 2 TT .. _ r l 2 V 2 2 



Fig. 105 a. 










































255 


Branches and Diversions. Art. 105 

/ 

where y is the difference in level between the reservoir surface 
and the surface of the water in a piezometer tube supposed to be 
inserted at the junction. This y is the friction-head consumed in 
the flow in the large main, and hence from formula ( 90 ) its value is 


Inserting this in the two equations, and placing for the velocities 
their values in terms of the discharges, they become 

"ff »■-/>+*>■+/■> 

from which the values of q x and q 2 are best obtained by trial. 

When it is required to determine the diameters from the given 
lengths, heads, and discharges, there are three unknown quan¬ 
tities, d , dij d 2 , to be found from only two equations, and the prob¬ 
lem is indeterminate. If, however, d be assumed, values of d x 
and d 2 may be found; and as d may be taken at pleasure, it ap¬ 
pears that an infinite number of solutions is possible. Another 
way is to assume a value of y, corresponding to a proper pressure- 
head at the junction; then the diameters are directly found from 
formula (97)3 for long pipes, in which h is replaced by y for the 
large main, and by H x — y and II 2 — y for the two branches. 

When two reservoirs, A\ and A 2 , are at a higher elevation than 
a third one into which they are to deliver water by pipes of length 
li and l 2 , both of which connect with a third pipe of length / which 
leads to the third reservoir, the above formulas also apply. In 
this case H x and Il 2 are the heights of the water levels in the reser¬ 
voirs A\ and A 2 above that in the third reservoir. 

When the principal main of a water-supply system enters 
a town, it divides into branches which deliver the water to different 
districts, and when such branches connect again with the princi¬ 
pal main, they form what may be called “diversions." Figure 
1056 shows a simple case, A being the reservoir and AB the prin¬ 
cipal main, while the pipe lines BCE and BDE form two routes 







256 Chap. 8. Flow of Water through Pipes 

or diversions through which water can flow to F. Let the main 
AB have the length l and the diameter d , the line BCE the length 
li and the diameter di, the line BDE the length l 2 and the diameter 
d 2 , while the line EF has the length l 3 and the diameter d 3 . Sup¬ 
pose that no water is drawn from the pipes except at F and be¬ 
yond, that the pressure-head Ff at F is /z 3 , and that the static 
head Ffi on F is h, and let it be required to find the velocity and 
discharge for each of the pipes. The total head H lost in friction 
is h — h 3 , and if W, W\, W 2 , and IF 3 represent the weights of water 



that pass any sections of the four pipes per second, the theorem 
of energy, neglecting the entrance head at A and the velocity-head 
at F, gives 

WH =W} l -—+W l fF'^+ W 2 f 2 + W r 3 k 

d 2 g di 2 g ' d 2 2 g dz 2 g 


Now referring to the figure where piezometers are shown on the 
profile at B and E it is seen that the loss of head in friction is the 
same for the diversions BCE and BDE ; accordingly there must 
exist the condition 7 9 , „ 

/ *1 _ r C ^2 

Jl-. ' ~ J 2 - 

di 2 g d 2 2 g 


and since W equals W x + W 2 and also equals W 3 , the above 
energy equation reduces to the simple form 



The values of vi and v 3 in terms of v are now to be inserted in this 
equation in order to determine v. From the conditions of con- 
























Branches and Diversions. Art. 105 


257 


tinuity of flow and that of equality of friction-head in the diver¬ 
sions, are found three equations. 


d 3 2 v 3 — d 2 v di 2 V\ + d 2 2 v 2 = d 2 v Vi V/di/di = v 2 V f 2 l 2 /d 2 

and accordingly, if the square roots of the quantities fih/di and 
SH<h be called e\ and e 2 for the sake of abbreviation, 


d 2 


Vo = 


e x d 2 


eodi 2 -f- e\do 2 
The above formula for H then reduces to 


Vi 


eod 2 


e 2 di 2 T e\do 2 


v 



1 fk ( d 2 V . r h( d Y ~1 9 

f d +h Z h dXe^ + e^) +h dXd)f 


from which v can be computed. Then vi, v 2 , and v 3 may be found, 
as also the discharges q, q h q 2 , and q 3 . 

As a numerical example, let / = io ooo, l\ = 2200, l 2 = 2800, 
lz = 1200 feet, and d = 12, d\ — 8, d 2 = 10, d 3 = 10 inches; let 
F be 184 feet below the water level in the reservoir and let the 
required pressure-head at F be 155 feet, so that H = 29 feet. 
Taking for the friction factors the mean value 0.02 (Art. 90 ), the 
value of fl/d is 200, that of fih/di is 66, that of f 2 l 2 /d 2 is 67.2, and 
that of fzh/dz is 28.8. The value of e\ is then 8.12 and that of e 2 is 
8.20, while d/dz is 1.2. Inserting these in the last formula, there 
is found v = 2.45 feet per second; then Vi = 2.16, v 2 = 2.14, 
and v 3 = 3.53 feet per second. As a check on these results the 
friction-heads for the four pipes may be computed, and these are 
found to be 18.6 feet for /, 4.8 feet for h and l 2 , and 5.5 feet for / 3 ; 
the sum of these is 28.9 feet, which is a sufficiently close agreement 
with the given 29.0 feet for a preliminary computation. The dis¬ 
charges are q = q 3 = 1.93, q\ =0.75, q 2 = 1.18 cubic feet per 
second, and the sum of qi and q 2 equals q , as should be the case. 
The computation may now be repeated, if thought necessary, 
the above velocities being used to take better values of the 
friction factors from Table 90 u. 


There are marked analogies between the flow of water in pipes 
and the flow of electricity in metallic conductors. Thus in Fig. 105 &, 
let BCE and BDE be two wires that carry the electric current passing 









258 


Chap. 8. Flow of Water through Pipes 


from A to F. If C x and C 2 be the currents in these circuits and R x 
and R 2 the resistances of the wires, it is an electric law that R\C X = 
R 2 C 2 , or the currents are inversely as the resistances. For water 
the discharges q x and q 2 are analogous to the electric currents, and, 
from the above equation, which expresses the equality of the friction- 
heads, it is seen that 

(/i/1 d \) 1 q\ = (fih/dz )' 1 q 2 


and accordingly the same law holds if the coefficients of q x and q - 2 be 
called resistances. If there be a third diversion BGE of length / 4 
and diameter d 4 connecting B and E, the current or the discharge 
through AB divides between the three diversions according to the 
same law, and , 2 7 o 7 2 

’ , h W _ r l\ V\ Z _ r h 

u -— -717 — -72 7 — 

d 4 2 g (ii 2 g d 2 2 g 


from which it is seen that (/ 4 / 4 /d 4 5 )^ 4 is equal to each of the corre¬ 
sponding expressions for the other diversions. This subject will re¬ 
ceive further discussion in Art. 208 . 


Prob. 105 . From a reservoir A a pipe 10000 feet long and 16 inches 
in diameter runs to a point B from which two diversions lead to E. The 
diversion BCE is 1600 feet long and 10 inches in diameter, while BDE con¬ 
sists of 2000 feet of 10-inch pipe and 1500 feet of 8-inch pipe. From the 
junction E, a pipe EF, 1000 feet long and 12 inches in diameter, leads to the 
business section of the town, where it is desired to have four fire streams 
deliver a total discharge of 900 gallons per minute through four hose lines of 
2^-inch smooth rubber-lined hose and 1 |-inch smooth nozzles. The point F is 
180 feet below the water level in the reservoir. Compute the velocity and 
discharge for each pipe and hose line, the friction-head lost in each and the 
pressure-head at the end F. 


Art. 106 . Cast-iron Pipes 

Cast-iron pipes generally range in size from 4 inches to 60 
inches in diameter the larger sizes being usually made to order. 
They are cast in 12-foot lengths and dipped into a hot bath of 
coal-tar. The joints are of the bell and spigot type, the space 
about the spigot being filled with lead or other material so as to 
form a tight joint. 

Some waters act rapidly on cast-iron causing the formation 
of tubercules of iron rust to such an extent that in the course of 



Cast-iron Pipes. Art. 106 


259 


years the diameter of the pipe may be reduced by fully 50 percent. 
Various machines have been devised for removing such incrusta¬ 
tions and deposits by scraping and thus in part restoring the orig¬ 
inal capacity of the pipe. No definite rule can be laid down for the 
selection of a proper friction factor for use in the design of a pipe. 
Each particular case must be carefully studied and the proper 
factor determined upon. Many experiments have been made in 
order to determine the friction factor in clean cast-iron pipes, and 
the results are tabulated in Table 90a. Other experiments have 
been made on pipes of various ages and a few of the results are 
here given in Table 106 in order to illustrate the range which 
is to be expected in the values of the friction factor. 


Table 106. Actual Friction Factors for Cast-iron Pipes 


Diam¬ 
eter in 

Age in 

Velocity in Feet per Second 

Reference 

Inches 

Years 

2.0 

3-0 

4.0 

12 

0 

0.021 

O.OI9 

O.Ol8 

Trans. Am. Soc. C. E., 

vol. 47 

12 

15 

O.076 


— 

Hering’s Kutter * 


12 

22 

0.12 I 

0.127 

— 

Hering’s Kutter * 


20 

5 

O.OI9 

0.022 

—■ 

Trans. Am. Soc. C. E., 

vol. 35 

20 

22 

O.069 

O.07I 

O.074 

Hering’s Kutter * 


36 

I? 



O.OI5 

Trans. Am. Soc. C. E., 

vol. 44 

36 

3$ 



0.059 

Trans. Am. Soc. C. E., 

vol. 44 

48 

0 



O.OI3 

Trans. Am. Soc. C. E., 

vol. 35 

48 

7 

0.028 


— 

Trans. Am. Soc. C. E., 

vol. 28 

48 

16 

0.023 

O.O23 

O.O23 

Trans. Am. Soc. C. E., 

vol. 35 


An inspection of the foregoing table indicates the great range 
in the values of the friction factor which are caused by progressive 
deterioration of the interior surface of a cast-iron pipe. Due 
allowance for this increase of the friction factor with age must 
be made in designing pipe lines and water mains. 

Prob. 106 . Compare the discharge of a new cast-iron pipe 20 inches 
in diameter and 10000 feet long under a head of 100 feet with that of the 
same pipe when 25 years old. 

* Hering and Trautwine’s translation of Gauguillet and Kutter’s Flow of 
Water in Rivers and Other Channels, New York, i88q, p. 155. 















260 


Chap. 8. Flow of Water through Pipes 


Art. 107 . Riveted Pipes 

Pipes 36 inches and larger in diameter have been made of 
wrought-iron or steel plates riveted together. Wrought-iron, 
however, is now but little used, on account of its higher cost, 
except in the form of thin sheets for temporary pipes. Each 
section usually consists of a single plate, which is bent into the 
circular form and the edges united by a longitudinal riveted lap 
joint. The different sections are then riveted together in trans¬ 
verse joints so as to form a continuous pipe. At AB (Fig. 107 a) 
is shown the so-called taper joint, where the end of each section 



Fig. 107a. 


goes into the end of the following one, as in a stovepipe, the flow 
occurring in the direction from A to B. At CD is seen the method 
of cylinder joints where the sections are alternately larger and 
smaller. For the large sizes double rows of rivets are used both 
in the longitudinal and transverse joints, the style of riveted 
joint depending on the pressure of water to be carried by the pipe. 
Riveted pipes have also been built with butt joints on both 
longitudinal and transverse seams, lap plates being on the outside. 

Pipes of this kind have long been in use in California in tem¬ 
porary mining operations, the diameters being from 0.5 to 1.5 
feet. In 1876 one was laid at Rochester, N.Y., partly 2 and 
partly 3 feet in diameter. Since 1892 several lines of large diam¬ 
eter have been constructed, notably the East Jersey pipe of 
3, 3.5 and 4 feet diameter, the Allegheny pipe of 5 feet diameter, 
and the Ogden and Jersey City pipes of 6 feet diameter. The 
steel pipe siphons now under construction on the Catskill Aque¬ 
duct for the city of New York vary in diameter from 9.5 to 11.2 
feet. These pipes will be covered with concrete as a protec¬ 
tion against exterior corrosion and will be lined inside with 2 
inches of Portland cement mortar both as a protective coating, 
as well as for the purpose of increasing their capacity. This, it 



















Riveted Pipes. Art. 107 


261 


may be noted, is a re-adoption of the old cement-lined pipe and 
it may be stated that the capacity of a pipe so lined is about 25 
percent greater than that of the same pipe without such lining. 


Owing to the friction caused by the rivets and joints the dis¬ 
charge from riveted pipes is less than that from cast-iron pipes 
in which the obstruction caused by the joints is very slight. 
The following values of the friction factor /, which have been 
derived from the data given by Herschel,* are applicable to new 
clean riveted pipes coated with asphaltum in the usual manner. 


Velocity in feet per second, v = 1 

13 ft. diam., / = 0.035 
[4 ft. diam., / = 0.025 
13! ft. diam.,/ = 0.025 


Cylinder joints 
Taper joints 


[4 ft. diam., / = 0.027 


2 

3 

4 

5 

6 

0.029 

0.024 

0.021 

0.019 

0.017 

0.022 

0.020 

0.020 

0.021 

0.021 

0.024 

0.023 

0.022 

0.022 

0.022 

0.026 

0.025 

0.024 

0.023 

0.023 


These friction factors are approximately double those given 
for new cast-iron pipes in Art. 90, this increase being largely due 
to the friction of the rivet heads and lapped joints though some 
of it is probably chargeable to the roughness of the asphaltum 
coating. It must be noted that these factors increase with age, 
thus when four years old the upper end of the above 4-foot 
cylinder joint pipe gave the following values: 

Velocity in feet per second, v = 1 2 3 4 5 6 

Cylinder joint 4 ft. diam., 7 = 0.042 0.032 0.030 0.029 0.029 0.029 

while the lower portion of this same pipe gave the following values: 

Velocity in feet per second, v= 1 2 3 4 5 6 

Cylinder joint 4 ft. diam., / = 0.027 0.024 0.023 0.024 0.024 0.024 

The diminution in capacity here shown during a period of 
4 years is greater for the upper than for the lower part of the line 
and this is to be ascribed in part at least to the greater number 
of vegetable growths which occur in most lines near, and for some 
distance below their intakes. 

When this same pipe was 15 years old (Art. 121 ) the values 
of the friction factor for its upper end were as follows: 

Velocity in feet per second, v = 1 2 3 4 5 

Cylinder joint 4 ft. diam., f— 0.036 0.036 


*115 Experiments on the Carrying Capacity of Large, Riveted, Metal Con¬ 
duits, New York, 1897. 


262 


Chap. 8. Flow of Water through Pipes 


and at this same age the values for its lower end were: 

Velocity in feet per second, v = i 2 3 4 5 6 

Cylinder joint 4 ft. diam., 7 = 0.046 0.034 0.032 0.031 

Similarly the 3§-foot-diameter taper joint pipe above referred 
to, when n years old, gave the following values for the friction 
factor: 

Velocity in feet per second, v= 1 2 3 4 5 6 

Taper joint 3J ft. diam., / = 0.050 0.036 0.034 0.032 

Experiments on the 6-foot Jersey City Water Supply Company * 
taper joint pipe gave the following values for the friction factor 


ages of 2 

months to 

5I years: 




Velocity in feet per second, 

V = I 

2 

3 

4 

at 

i year, 

/ = 0.021 

0.022 

0.022 

0.022 

at 

U years, 

/ = 0.029 

0.026 

0.026 

0.025 

at 

2I years, 

7=0.034 

0.029 

0.027 

0.027 

at 

5! years, 

7= 0.036 

0.034 

0-035 



Gagings by Marx, Wing, and Hoskins j of the flow through a 
steel riveted pipe 6 feet in diameter with butt joints when new, 
and again after two years’ use furnish the following values of the 
friction factor /: 

Velocity in feet per second, v = 1 2 3 4 5 6 

1897, f = 0.021 0.021 0.022 0.021 

1899, f = 0.038 0.027 0.025 0.024 0.023 0.023 

These results indicate a marked diminution with age in carry¬ 

ing capacity. This reduction is in part due to the formation of 
blisters in the asphaltum coating, which is generally used, in part, 
to the formation of tubercules or rust spots and in part to vegeta¬ 
ble growths and incrustations formed by deposits from the water. 

The so-called lock-bar pipe (Fig. 1076 ) was first used on the Cool- 
gardie line in Australia and since 1900 has been introduced to a con¬ 
siderable extent in the United States. In this style of pipe the transverse 
joints are made up with rivets, as in the ordinary riveted pipe, but the 


* Here published by courtesy of Jersey City Water Supply Company, 
f Transactions American Society of Civil Engineers, 1898, vol. 40, 
p. 471; and 1900, vol. 44, P- 34 - 


Wood Pipes. Art. 108 


263 


longitudinal joints are made by clamping the edges of the plates under 
heavy pressure into a grooved bar which thus holds them together and 
makes a joint of exceptional strength. No longitudinal rivets there¬ 
fore interfere with the flow, and 
as the plates of which the pipe M/// 

is made can be used with their 
longer edges parallel to the axis 
of the pipe, the number of 
transverse joints can be reduced 



Fig. 1076 . 


from 50 to 60 per cent. The carrying capacity of this style of pipe 
is probably materially in excess of that of riveted pipe, but no re¬ 
corded experiments are available from which values of the friction 
factor can be stated. 


Prob. 107 . Construct curves showing the progressive increase with age 
in the value of the friction factor/ for riveted steel pipes of 42, 48, and 60 
inches in diameter. 


Art. 108 . Wood Pipes 

Wood pipes were used in several American cities during the 
years 1750-1850, these being made of logs laid end to end, a 3 or 
4 inch hole having been first bored through each log. Pipes 
formed of redwood staves were first used in California about 
1880, these staves being held in place by bands of wrought-iron 
arranged so that they could be tightened by a nut and screw. 
Several long lines of these large conduit pipes have been built 
in the Rocky mountains and Pacific states. They have also 
been used there for city mains to a limited extent and recently 
have been introduced in the East on main distributing lines. 

Gagings of a wood pipe 6 feet in diameter were made by Marx, 
Wing, and Hoskins, in connection with those of the steel pipe 
cited in Art. 107 . The values of the friction factor/ deduced from 
their results for velocities ranging from 1 to 5 feet per second are 

Velocity in feet per second, v = 1 2 3 4 5 

1897, / = 0.026 0.019 0.017 0.016 

1899, / = 0.019 0.018 0.017 0.017 0.017 

These show that this wood pipe became smoother after two years’ 

use, while the steel pipe became rougher. 



264 


Chap. 8. Flow of Water through Pipes 


T. A. Noble’s gagings of wood pipes 3.67 and 4.51 feet in 
diameter furnish similar values of /.* For the smaller pipe / 
ranges from 0.021 to 0.019, with velocities ranging from 3.5 
to 4.8 feet per second. For the larger pipe / ranges from 0.019 
to 0.016, with velocities ranging from 2.3 to 4.7 feet per second. 
From Adams’ measurements on a pipe 1.17 feet in diameter the 
values of / range from 0.027 to 0.020, with velocities ranging from 
0.7 to 1.5 feet per second. Noble’s discussion of all the recorded 
gagings on wood pipes show certain unexplained discrepancies, 
and he proposes special empirical formulas to be used for precise 
computations. Wooden stave pipes after being in service some 
time may undergo considerable alterations in form, as the circle 
is apt to be deformed into an ellipse. 

By the help of the formulas of the preceding pages, computations 
for the velocity and discharge of steel and wood pipes under given heads 
may be readily made. As such pipes are generally long, the formulas 
of Art. 97 will usually apply. In designing a pipe line a liberal factor 
of safety should be introduced by taking a value of / sufficiently large 
so that the discharge may not be found deficient after a few years’ 
use has deteriorated its surface. 

Prob. 108 . What is the discharge, in gallons per day, of a wood stave 
pipe 5 feet in diameter when the slope of the hydraulic gradient is 47.5 feet 
per mile ? 

Art. 109 . Fire Hose 

Fire hose is generally 2J inches in diameter, and lined with 
rubber to reduce the frictional losses. The following values 
of the friction factor / have been deduced from the experiments 
of Freeman.f 


Velocity in feet per second, 

v = 

4 

6 

10 

15 

20 

Unlined linen hose, 

/ = 

0.038 

0.038 

0.037 

0-035 

0.034 

Rough rubber-lined cotton, 

/ = 

0.030 

0.031 

0.031 

0.030 

0.029 

Smooth rubber-lined cotton, 

/ = 

0.024 

0.023 

0.022 

0.019 

0.018 

Discharge, gallons per minute = 

61 

92 

153 

230 

306 


By the help of this table computations may be made on flow of 
water through fire hose in the same manner as for pipes. It is 

* Transactions American Society of Civil Engineers, 1902, vol. 49, pp. 112, 143. 
f Transactions American Society of Civil Engineers, 1889, vol. 21, p. 303; 346. 


Fire Hose. Art. 109 


265 


seen that the friction factors for the best hose are slightly less than 
those given for 2^-inch pipes in Table 90 a. 

When the hose line runs from a steamer to the nozzle, instead 
of from a reservoir, the head h is that due to the pressure p at 
the steamer pump (Art. 11 ). If this hose line is of uniform diam¬ 
eter the velocity in the hose and nozzle may be computed by 
Art. 101 and the discharge is then readily found. For example, 
let the hose be 2\ inches in diameter and 400 feet long, the pres¬ 
sure at the steamer be 100 pounds per square inch, which corre¬ 
sponds to a head of 230.4 feet, and the nozzle be 1J inches in diam¬ 
eter with a coefficient of velocity of 0.98. Then, neglecting 
the loss of head at entrance, and using for / the value 0.03, the 
velocity from the nozzle is found to be 66.0 feet per second, which 
gives a velocity-head of 67.7 feet and a discharge of 180 gallons 
per minute. The head lost in friction is 230.4 — 67.7 = 162.7 
feet, of which 2.8 feet are lost in the nozzle and the remainder 
in the hose. 

Sometimes the hose near the steamer is larger in diameter 
than the remaining length. Let 4 be the length and d\ the di¬ 
ameter of the larger hose, and 4 and d 2 the same quantities for 
the smaller hose. Let C\ be the coefficient of velocity for a smooth 
nozzle, D its diameter, and V the velocity of the stream issuing 
from the nozzle. By reasoning as in Arts. 93 and 101 , and neg¬ 
lecting losses of head at entrance and in curvature, there is 
found for the velocity at the end of the nozzle 



and the discharge is given by q = \ r rrD 2 V. For example, let /z = 
230.4, /i = ioo, 4 = 300 feet; di = 3, <4 = 2.5, D = 1.125 inches; 
Ci =0.98, and fi = / 2 = 0.03. Then, by the formula F = 69.7 feet per 
second, which gives a velocity-head of 75.5 feet and a discharge 
of 190 gallons per minute. This example is the same as that 
of the preceding paragraph, except that a larger hose is used 
for one-fourth of the length, and it is seen that its effect is to 
increase the velocity-head nearly 12 per cent and the discharge 




266 


Chap. 8. Flow of Water through Pipes 


nearly 6 per cent. For this case the head lost in friction is 154.9 
feet, of which 3.1 feet are lost in the nozzle and the remainder 
in the 400 feet of hose. 

In using the above formula the tip of the nozzle is supposed to be 
on the same level with the pressure gage at the steamer pump and the 
head h is given in feet by 2.304 p, where p is the gage reading in pounds 
per square inch. When the tip of the nozzle is a vertical distance z 
above this gage, h is to be replaced by h — z in the formula; when it 
is the same vertical distance below the gage, h is to be replaced by 
h -f- z. In the former case gravity decreases and in the latter case 
it increases the velocity and discharge. The above formula applies 
also to the case of a hose connected to a hydrant, if h is the effective- 
head at the entrance, that is, the pressure-head plus the velocity-head 
iii the hydrant. In Art. 201 will be found further discussions re¬ 
garding pumping through fire hose. 

At a hydrant of diameter d x the pressure-head is h x . To this is 
attached a hose of length l and diameter d and to the end of the hose 
a nozzle of diameter D and velocity coefficient c x . Neglecting losses 
at entrance and in curvature the formula for computing the velocity 
of the jet issuing from the nozzle, when its tip is held at the same level 
as the gage that indicates the pressure-head, is 



Prob. 109 . When the pressure-gage at the steamer indicates 83 pounds 
per square inch, a gage on the leather hose 800 feet distant reads 25 pounds. 
Compute the value of the friction factor /, the discharge per minute being 
121 gallons. If the second gage be at the entrance to a 1 pinch nozzle, 
compute its coefficient of velocity. 

Art. 110 . Other Formulas for Flow in Pipes 

The formulas thus far presented in this chapter are based 
upon the assumption that all losses of head vary with the square 
of the velocity. This is closely the case for the velocities common 
in engineering practice, but for velocities smaller than 0.5 feet 
per second the losses of head due to friction have been found to 
vary at a less rapid rate, and in fact nearly as the first power of 




Other Formulas for Flow in Pipes. Art. 110 


267 


the velocity. Probably at usual velocities the loss of head in 
friction is composed of two parts, a small part varying directly 
with the velocity which is due to cohesive resistance along the 
surface, and a large part varying as the square of the velocity 
which is due to impact as illustrated in Fig. 90 . This was recog¬ 
nized by the early hydraulicians who, after defining the friction 
head and friction factor as in ( 90 ), by the formula 



endeavored to express / in terms of the velocity v. 
D’Aubisson deduced 


/ 


0.0269 + 


0.00484 

v 


Thus, 


and Weisbach advocated the form 


. , 0.0172 

/ = 0.0144 H- r 

Darcy, on the other hand, expressed / in terms of d, namely, 

. . 0.00167 

/ = 0.0199 H-—- L 


All these expressions are for English measures, v being in feet 
per second and d in feet. Later investigations show, however, 
that / varies with both v and d, and the best that can now be 
done is to tabulate its values as in Table 90 a. In fact it may 
be said that the theory of the flow of water in pipes at common 
velocities is not yet well understood. 

Many attempts have been made to express the velocity of 
flow in a long pipe by an equation of the form 

v = « ■ d^ih/iy 

in which «, 0 , and 7 are to be determined from experiments in 
which v, d, h, and l have been measured. The exponential for¬ 
mula deduced by Lampe for clean cast-iron pipes varying in 
diameter from one to two feet is 


v = 77.7 d om (h/l)°‘ br ° 5 


(HO) 





268 Chap. 8. Flow of Water through Pipes 

in which d, h , and l are to be taken in feet, and v will be found in 
feet per second. From this are derived 

q = 61.0 d 2,694 (V0 0 ' 555 d = 0.217 

by which discharge and diameter may be computed. Other 
investigators find different values of 0 and 7, the values /3 = f 
and 7 = | being frequently advocated. 

The formula of Chezy (Art. 113 ), that of Kutter (Art. 118 ), 
that of Bazin (Art. 122 ), and that of Williams and Hazen (Art. 
124 ), are often used for long pipes, care being taken to select the 
proper value of c for the first, of n for the second, of m for the 
third, and of c for the fourth. The formulas of Kutter and Bazin 
are sometimes more advantageous than the others since in using 
them the roughness of the surface of the pipe can better be 
taken into account. 

The formulas of this chapter do not apply to very small pipes 
and very low velocities, and it is well known that for such condi¬ 
tions the loss of head in friction varies as the first power of the velocity. 
This was shown in 1843 by Poiseuille, who made experiments in order 
to study the phenomena of the flow of blood in veins and arteries. 
For pipes of less than 0.03 inches diameter he found the head h to 
be given by h = C\lv/d 2 where C\ is a constant factor for a given tem¬ 
perature, v is the velocity, d the diameter, and l the length of the pipe. 
Later researches indicate that the laws expressed by this equation 
also hold for large pipes provided the velocity be very small, and that 
there is a certain critical velocity at which the law changes and beyond 
which h = C 2 lv 2 /d, as for the common cases in engineering practice. 
This critical point appears to be that where the filaments cease to move 
in parallel lines and where the impact disturbances illustrated in Fig. 
90 begin. For a very small pipe the velocity may be high before this 
critical point is reached ; for a large pipe it happens at very low veloci¬ 
ties. Experiments devised by Reynolds enable the impact disturb¬ 
ance to be actually seen as the critical velocity is passed, so that its 
existence is beyond question. It may also be noted that the velocity 
of flow through a submerged sand filter bed varies directly as the first 
power of the effective head. 

Prob. 110 . Solve Problems 94 and 95 by the use of the above formulas 
of Lampe. _ . 






Computations in Metric Measures. Art. Ill 


269 


Art. 111 . Computations in Metric Measures 

Nearly all the formulas of this chapter are rational in form, the 
coefficient of velocity c h the factors / and f h and the factors m, m iy 
m 2 , and m! are abstract numbers which have the same values in all 
systems of measures. 

(Art. 90 ) The mean value of the friction factor/ is 0.02, and Table 
111 a gives closer values corresponding to metric arguments. For 


Table 111a. Friction Factors for Clean Iron Pipes 

Arguments in Metric Measures 


Diameter in 

Velocity in Meters per Second 

Centimeters 

0.3 

0.6 

1.0 

1-5 

2-5 

4-5 

1-5 

O.047 

O.O4I 

O.036 

O.033 

O.O3O 

0.028 

3 - 

.038 

.032 

.030 

.027 

.025 

.023 

8. 

.031 

.028 

.026 

.024 

.023 

.021 

16. 

.027 

.026 

.025 

.023 

.021 

.019 

30. 

.025 

.024 

.023 

.021 

.019 

.017 

40. 

.024 

.023 

.022 

.019 

.Ol8 

.Ol6 

60. 

.022 

.020 

.019 

.017 

.015 

.013 

90. 

.019 

.Ol8 

.Ol6 

.015 

.013 

.012 

120. 

.017 

.Ol6 

.015 

.013 

.012 


180. 

.015 

.014 

.013 

.012 




example, let l = 3000 meters, d = 30 centimeters = 0.3 meters, and 
v = 1.75 meters per second. Then from the table / is 0.022, and 

h" = 0.022 X X = 34-3 meters, 

0.3 19.6 

which is the probable loss of head in friction. By the use of Table 
1116 approximate computations may be made more rapidly, thus for 
this case the loss of head for 100 meters of pipe is found to be 1.10 
meters, hence for 3000 meters the loss of head is 33 meters. 

(Art. 94 ) The metric value of \ r rr^/ 2 g is 3.477 and that of 8/7r 2 g 
is 0.2653. 

(Art. 95 ) When ( 95 ) is used in the metric system, the constant 
0.4789 is to be replaced by 0.6075 ; here q is to be in cubic meters per 

second, and l and d in meters. 

1 





















270 


Chap. 8. Flow of Water through Pipes 


Table 1116 . Friction Head for ioo Meters of Clean Iron 

Pipe 

Metric Measures 


Diameter in 

Velocity in Meters per Second 

Centimeters 

0.3 

0.6 

1.0 

, s 

2.5 

4-5 


Meters 

Meters 

Meters 

Meters 

Meters 

Meters 

1-5 

I.44 

5.02 

12.2 




3 - 

0.58 

1.96 

5 - 1 ° 

10.3 

26.6 


8. 

.18 

0.64 

1.66 

3-45 

9-23 

27.I 

16. 

.08 

•30 

.80 

I.65 

4.09 

12.3 

30 - 

.04 

•IS 

•39 

0.80 

2.02 

5.85 

40. 

•03 

.10 

.28 

•54 

1-43 

4-13 

60. 

.02 

.06 

.16 

•33 

0.80 

2.24 

90. 

.OI 

.04 

.09 

.19 

.46 

1.38 

120. 


.02 

.06 

.12 

•32 


180. 


.OI 

.04 

.08 




(Art. 97 ) In ( 97 ) 2 the two constants are 4.43 and 3.48 instead 
of 8.02 and 6.30. In ( 97 ) 3 the constant is 0.607 instead of 0.479. 

(Arts. 106 , 107 , and 108 ) The friction factors / for cast iron, steel 
and wood pipes may be taken for metric arguments by using the 
velocities in meters per second, namely, by writing 0.3, 0.6, 0.9, 1.2, 
1.5, 1.8 meters per second, instead of 1, 2, 3, 4, 5, 6 feet per second. 

(Art. 109 ) For fire hose the values of the friction factor / for 
metric data are as follows, for hose 6.35 centimeters in diameter: 


Velocity, meters per second, 

v = 

1.22 

1.83 

3-05 

4-57 

6.10 

Unlined linen hose, 

/ = 

0.038 

0.038 

0.037 

0.035 

0.034 

Rough rubber-lined cotton, 

/ = 

0.030 

0.031 

0.031 

0.030 

0.029 

Smooth rubber-lined cotton, 

/ = 

0.024 

0.023 

0.022 

0.019 

0.018 

Discharge, liters per minute, 

= 

231 

348 

579 

M 

00 

1158 


(Art. 110 ) In the metric system the formulas for the friction 
factor / are the same as those in the text, except that the numerator 
of the last term is to be divided by 3.28 in the formulas of D ’Aubisson 
and Darcy and by 1.81 in that of Weisbach. Lampe’s formula is 

v = 54.1 d° 694 (Zr/ 0 0,555 

and his formulas for discharge and diameter are 


q = 42.5 d 2 - m (h/l )° 555 


d= 0.249 g 0 371 (VO° 206 



























Computations in Metric Measures. Art. Ill 


271 


in which d , /;, and I are in meters, v in meters per second, and q in 
cubic meters per second. 

Prob. 110(7. Compute the diameter, in centimeters, for a pipe to de¬ 
liver 500 liters per minute under a head of 2 meters, when its length is 100 
meters. Also when the length is 1000 meters. 

Prob. 1106 . Compute the velocity-head and discharge for a pipe 1 meter 
in diameter and 856 meters long under a head of 64 meters. Compute the 
same quantities when a smooth nozzle 5 centimeters in diameter is attached 
to the end of the pipe. 

Prob. 110 c. A compound pipe has the three diameters 15, 20, and 30 
centimeters, the lengths of which are 150, 600, and 430 meters. Compute 
the discharge under a head of 16 meters. 

Prob. 110 c/. A steel-riveted pipe 1.5 meters in diameter is 7500 meters 
long. Compute the velocity and discharge under a head of 30.5 meters. 

Prob. 110 c. The value of C1 in Poiseuille’s formula for small pipes is 
0.0000177 f° r English measures at io° centigrade. Show that its value is 
0.0000690 for metric measures. 

Prob. 110 /. In Fig. 1056 let the pipe AB be 3000 meters long and 30 
centimeters in diameter, BCD be 800 meters long and 20 centimeters in diam¬ 
eter, BCE be 1000 feet long and 20 centimeters in diameter, and EF be 
300 meters long and 30 centimeters in diameter. Compute the velocity and 
discharge for each pipe when the total lost head H is 12.5 meters. 


272 


Chap. 9 . Flow in Conduits and Canals 


CHAPTER 9 

FLOW IN CONDUITS AND CANALS 
Art. 112 . Definitions 

From the earliest times water has been conveyed from place 
to place in artificial channels, such as troughs, aqueducts, ditches, 
and canals, there being no head to cause the flow except that due 
to the slope. The Roman aqueducts were usually rectangular 
channels about 2\ feet wide and 5 feet deep, lined with cement, 
sometimes running underground and sometimes supported on 
arches. The word “conduit” will be used as a general term for 
a channel of any shape lined with timber, mortar, or masonry, 
and will also include large metal pipes, troughs, and sewers. 
Conduits may be either open, as in the case of troughs, or closed, 
as in sewers and most aqueducts. Ditches and canals are con¬ 
duits in earth without artificial lining. Most of the principles 
relating to conduits and canals apply also to streams, and the 
word “ channel ” will be used as applicable to all cases. 

The wetted perimeter of the cross-section of a channel is 
that part of its boundary which is in contact with the water. 
Thus, if a circular sewer of diameter d be half full of water, the 
wetted perimeter is \nrd. In this chapter the letter p will desig¬ 
nate the wetted perimeter. 

The hydraulic radius of a water cross r section is its area divided 
by its wetted perimeter, and the letter r will be used to designate 
it. If a is the area of the cross-section, the hydraulic radius of 
that section is found by 

r = a/p 

The letter r is of frequent occurrence in formulas for the flow 
in channels; it is a linear quantity which is always expressed in 
the same unit as p , and hence its numerical value is different in 


Definitions. Art. 112 


273 


different systems of measures. It is frequently called the hy¬ 
draulic depth or hydraulic mean depth, because for a shallow 
section its value is but little 
less than the mean depth of 
the water. Thus, in Fig. 112 , 
if b be the breadth on the 
water surface, the mean depth is a/b, and the hydraulic radius is 
a /and these are nearly equal, since the length of p is but 
slightly larger than that of b. 

The hydraulic radius of a circular cross-section filled with 
water is one-fourth of the diameter; thus 

r = ajp = \ 7 rd 2 / 7 rd = \d 

The same value is also applicable to a circular section half filled 
with water, since then both area and wetted perimeter are one- 
half their former values. 

The slope of the water surface in the longitudinal section, 
designated by the letter s, is the ratio of the fall h to the length 
l in which that fall occurs, or 

5 = hi 

The slope is hence expressed as an abstract number, which is in¬ 
dependent of the system of measures employed. To determine 
its value with precision h must be obtained by referring the water 
level at each end of the line to a bench-mark by the help of a hook 
gage or other accurate means, the benches being connected by 
level lines run with care. The distance l is not measured hori¬ 
zontally but along the inclined channel, and it should be of con¬ 
siderable length in order that the relative error in h may not be 
large. If s = o there is no slope and no flow; but vdien there 
is even the smallest slope the force of gravity furnishes a com¬ 
ponent acting down the inclined surface, and motion ensues. 
The velocity of flow evidently increases with the slope. 

The flow in a channel is said to be steady when the same quan¬ 
tity of water per second passes through each cross-section. If 
an empty channel be filled by admitting water at its upper end, 
the flow is at first non-steady or variable, for more water passes 



Fig. 112 . 











274 Chap. 9 . Flow in Conduits and Canals 

through one of the upper sections per second than is delivered 
at the lower end. But after sufficient time has elapsed the flow 
becomes steady; when this occurs the mean velocities in different 
sections are inversely as their areas (Art. 31 ). 

Uniform flow is that particular case of steady flow where all 
the water cross-sections are equal, and the slope of the water 
surface is parallel to that of the bed of the channel. If the sec¬ 
tions vary, the flow is said to be non-uniform, although the con¬ 
dition of steady flow is still fulfilled. In this chapter only the 
case of uniform flow will be discussed. 

The velocities of different filaments in a channel are not equal, 
as those near the wetted perimeter move slower than the central 
ones, owing to the retarding influence of friction. The mean of all 
the velocities of all the filaments in a cross-section is called the 
mean velocity v. Thus if v', v", etc., be velocities of different 
filaments, 

v' + v" + etc. 

v =- 

n 

in which n is the number of filaments. Let a be the area of 
the cross-section and let each filament have the small cross-section 
of area a then n — a/a ', and hence, 

av = a'(y' + v" -f etc.) 

But the second member is the discharge q; that is, the quantity 
of water passing the given cross-section in one second. There¬ 
fore the mean velocity may be also determined by the relation 

v = q/a 

The filaments which are here considered are in part imaginary, 
for experiments show that there is a constant sinuous motion of 
particles from one side of the channel to the other. The best 
definition for mean velocity hence is, that it is a velocity which 
multiplied by the area of the cross-section gives the discharge, 
or v — q/a. 

Prob. 112 . Compute the hydraulic radius of a rectangular trough 
whose width is 5.6 feet and depth 2.8 feet. 



Formula for Mean Velocity. Art. 113 


275 


Art. 113 . Formula for Mean Velocity 

When all the wetted cross-sections of a channel are equal, 
and the water is neither rising nor falling, having attained the 
condition of steady flow, the flow is said to be uniform. This is 
the case in a conduit or canal of constant size and slope whose 
supply does not vary. The same quantity of water per second 
then passes each cross-section, and consequently the mean veloc¬ 
ity in each section is the same. This uniformity of flow is due 
to the resistances along the interior surface of the channel, for 
were it perfectly smooth the force of gravity would cause the 
velocity to be accelerated. The entire energy of the water due 
to the fall h is hence expended in overcoming resistances caused 
by surface roughness. A part overcomes friction along the sur¬ 
face, but most of it is expended in eddies of the water, whereby 
impact results and heat is generated. A complete theoretic 
analysis of this complex case has not been perfected, but if the 
velocity be not small, the discussion given for pipes in Art. 90 
applies equally well to channels. 

Let W be the weight of water passing any cross-section in 
one second,/ 7 the force of friction per square unit along the surface, 
p the wetted perimeter, and h the fall in the length l. The poten¬ 
tial energy of the fall is Wh. The total resisting friction is Fpl, 
and the energy consumed per second is Fplv, if v be the velocity. 
Accordingly Fplv equals Wh. But the value of W is wav , if w 
is the weight of a cubic foot of water and a the area of the 
cross-section in square feet. Therefore Fpl = wall, and since 
a/p is the hydraulic radius r, and h/l is the slope s , this reduces 
to F = wrs, which is an approximate expression for the resisting 
force of friction on one square unit of the surface of the channel. 
In order to establish a formula for the mean velocity the value 
of F must be expressed in terms of v, and this can only be done 
by studying the results of experiments. These indicate that F 
is approximately proportional to the square of the mean velocity. 
Therefore if c is a constant, the mean velocity is 

V = C VrS 


( 113 ) 


276 


Chap. 9 . Flow in Conduits and Canals 


which is the formula first advocated by Chezy in 1775. This is 
really an empirical expression, since the relation between F and 
v is derived from experiments. The coefficient c varies with the 

roughness of the bed and with other circumstances. 

% 

Another method of establishing Chezy’s formula for channels 
is to consider that when a pipe on a uniform slope is not under 
pressure, the hydraulic gradient coincides with the water surface. 
Then formula ( 90 ) may be used by replacing h" by h and d by 
its value 4 r. Accordingly 

h = \ or v = V8g// V/\s 

r2g 

in which the quantity VSg/f is the Chezy coefficient. 

This coefficient c is different in different systems of measures 
since it depends upon g. For the English system it is found that 
c usually lies between 30 and 160, and that its value varies with 
the hydraulic radius and the slope, as well as with the roughness 
of the surface. To determine the value of c for a particular case 
the quantities v, r, and s are measured, and then c is computed. 
To find r and s linear measurements and leveling are required. 
To determine v the flow must be gaged either in a measuring 
vessel or by an orifice or weir, or, if the channel be large, by floats 
or other indirect methods described in the next chapter, and then 
the mean velocity v is computed from v = q/a. It being a matter 
of great importance to establish a satisfactory formula for mean 
velocity, thousands of such gagings have been made, and from 
the records of these the values of the coefficients given in the 
tables in the following articles have been deduced. 

Prob. 113 . Compute the value of c for a circular masonry conduit 
6 feet in diameter which delivers 65 cubic feet per second when running half 
full, its slope or grade being 1.5 feet in 1000 feet.. 

Art. 114 . Circular Conduits, Full or Half Full 

When a circular conduit of diameter d runs either full or half 
full of water, the hydraulic radius is \d, and the Chezy formula 
for mean velocity is 

v = c vVs = c • f Vds 




Circular Conduits, Full or Half Full. Art. 114 277 


The velocity can then be computed when c is known, and for 
this purpose Table 114 gives Hamilton Smith's values of c for 
pipes and conduits having quite smooth interior surfaces and no 
sharp bends.* The discharge per second then is 

q = av = c • \a V ds 

in which a is either the area of the circular cross-section or one- 
half that section, as the case may be. 


Table 114 . Coefficients c for Circular Conduits 


Diameter 

Velocity in Feet per Second 

in Feet 

1 

2 

3 

4 

L 6 

10 

15 

I. 

96 

104 

I09 

112 

116 

121 

124 

i -5 

103 

III 

Il6 

119 

123 

129 

132 

2. 

109 

Il6 

1 2 I 

I2 4 

129 

134 

138 

2-5 

113 

120 

125 

128 

133 

139 

143 

3 - 

117 

124 

128 

132 

136 

143 

147 

3-5 

120 

127 

131 

135 

139 

146 

151 

4 - 

123 

130 

134 

137 

142 

150 

155 

5 - 

128 

134 

139 

142 

147 

155 


6. 

132 

138 

142 

145 

150 



7 - 

135 

141 

145 

149 

153 



8. 

137 

143 

I48 

151 





To use Table 114 a tentative method must be employed 
since c depends upon the velocity of flow. For this purpose there 
may be taken roughly 

mean Chezy coefficient c = 125 

and then v may be computed for the given diameter and slope; 
a new value of c is then taken from the table and a new v com¬ 
puted; and thus, after two or three trials, the probable mean 
velocity of flow is obtained. The value of the diameter d must 
be expressed in feet. 

For example, let it be required to find the velocity and dis¬ 
charge of a semicircular conduit of 6 feet diameter when laid on 
a grade of 0.1 feet in 100 feet. First, 

fl = i25XjV6X 0.001 = 4.8 feet per second. 

* Hydraulics (London and New York, 1886), p. 271. 




















278 


Chap. 9 . Flow in Conduits and Canals 


For this velocity the table gives 147 for c ; hence 
v = 147 X \ V0.006 = 5.7 feet per second. 

Again, from the table c = 150, and 

v = 150 X | V0.006 = 5.8 feet per second. 

This shows that 150 is a little too large; for c = 149.5,^ is found 
to be 5.79 feet per second, which is the final result. The discharge 
per second now is 

q = 0.7854 X | X 36 X 5.79 = 81.9 cubic feet, 
which is the probable flow under the given conditions. 

To find the diameter of a circular conduit to discharge a given 
quantity under a given slope, the area a is to be expressed in terms 
of d in the above equation, which is then to be solved for d ; thus. 




the first being for a conduit running full and the second for one 
running half full. Here c may at first be taken as 125; then d 
is computed, the approximate velocity found from v = q/l^rd 2 , 
and with this value of v a value of c is selected from the table, 
and the computation for d is repeated. This process may be 
continued until the corresponding values of c and v are found to 
be in close agreement. 

As an example of the determination of diameter let it be re¬ 
quired to find d when q = 81.9 cubic feet per second, 5 = 0.001, 
and the conduit runs full. For c = 125 the formula gives d = 4.9 
feet, whence v = 4.37 feet per second. From the table c may be 
now taken as 142, and repeating the computation d = 4.64 feet, 
whence v = 4.84 feet per second, which requires no further 
change in the value of c. As the tabular coefficients are based 
upon quite smooth interior surfaces, such as occur only in new, 
clean, iron pipes, or with fine cement finish, it might be well to 
build the conduit 5 feet or 60 inches in diameter. It is seen 
from the previous example that a semicircular conduit of 6 feet 
diameter carries the same amount of water as is here carried by 
one of 4.64 feet diameter which runs entirely full. 






Circular Conduits, Partly Full. Art. 115 


279 


Circular conduits running full of water are long pipes and all 
the formulas and methods of Arts. 94 and 95 can be applied also 
to their discussion. From Art. 113 it is seen that 

c=V8g// or c= 16.04/ v 7 

in which/ is to be taken from Table 90 a. Values of c computed in 
this manner will not generally agree closely with the coefficients 
of Smith, partly because the values of / are given only to three 
decimal places, and partly because Table 90 a for pipes was con¬ 
structed from experiments on smoother surfaces than those of 
conduits. An agreement within 5 per cent in mean velocities de¬ 
duced by different methods is all that can generally be expected 
in conduit computations, and if the actual discharge agrees as 
closely as this with the computed discharge, the designer can 
be considered a fortunate man. 

All of the laws deduced in the last chapter regarding the relation 
between diameter and discharge, relative discharging capacity, 
etc., hence apply equally well to circular conduits which run either 
full or half full. If the conduit be full, it matters not whether it be 
laid truly to grade or whether it be under pressure, since in either case 
the slope 5 is the total fall h divided by the total length. Usually, 
however, the word “ conduit ” implies a uniform slope for considerable 
distances, and in this case the hydraulic gradient coincides with the 
surface of the flowing water. 

Prob. 114 . Find the diameter of a circular conduit to deliver when 
running full 16 500 000 gallons per day, its slope being 0.00016. 


Art. 115 . Circular Conduits, Partly Full 

Let a circular conduit with the slope 5 be partly full of water, 
its cross-section being a and hydraulic radius r. Then the mean 
velocity and the discharge are given by 

v = c~Vrs q = cay/rs 

The mean velocity is hence proportional to Vr and the discharge 
to a Vr, provided that c be a constant. Since, however, c varies 
slightly with r, this law of proportionality is only approximate. 



280 


Chap. 9 . Flow in Conduits and Canals 


When a circular conduit of diameter d runs either full or half 
full, its hydraulic radius is \d (Art. 112 ). If it is filled to the 

depth d' (Fig. 115 ), the wetted perimeter is 

p = j 7 rd + d arc sin •—-— 

and the sectional area of the water surface is 

Fig. 115 . a = \dp + (d! - %d) -Vd'(d-d') 

From these p and a can be computed, and then r is found by 
dividing a by p. Table 115 gives values of p, a , and r for a circle 
of diameter unity for different depths of water. To find from it 
the hydraulic radius for any other circle it is only necessary to 
multiply the tabular values of r by the given diameter d. The 
table shows that the greatest value of the hydraulic radius occurs 
when d' = o.Sid, and that it is but little less when d' = o.Sd. 
In the fifth and sixth columns of the table are given values of V/ 
and a Vr for different depths in the circle of diameter unity; 
these are approximately proportional to the velocity and discharge 
which occur in a circle of any size. The table shows that the 
greatest velocity occurs when the depth of the water is about eight- 



Table 115 . Cross-sections of Circular Conduits 


Depth 

a 

Wetted 

Perimeter 

P 

‘ Sectional 
Area 

a 

Hydraulic 

Radius 

r 

Velocity 

Discharge 
a \/ r 

Full i.o 

3.142 

0.7854 

O.25 

0.5 

0-393 

o -95 

2.691 

0.7708 

0.286 

0.535 

0.413 

0.9 

2.498 

0-7445 

0.298 

0.546 

0.406 

o.8i 

2.240 

0.6815 

0.3043 

0.552 

0.376 

o.8 

2.214 

0.6735 

O.3042 

0.552 

O.372 

0.7 

I.983 

0.5874 

0.296 

0.544 

0.320 

o.6 

I.772 

0.4920 

00 

CN 

d 

0.527 

0.259 

Half Full 0.5 

I- 57 I 

O.3927 

0.25 

0.5 

0.196 

0.4 

1.369 

O.2934 

0.214 

0.463 

0.136 

0.3 

1-159 

0.1981 

0.171 

0.414 

0.0820 

0.2 

0.927 

O.II18 

0.121 

0.348 

0.0389 

0.1 

0.643 

0.0408 • 

0.0635 

0.252 

0.0103 

Empty 0.0 

0.0 

0.0 

0.0 

0.0 

0.0 


































Circular Conduits, Partly Full. Art. 115 281 

tenths of the diameter, and that the greatest discharge occurs 
when the depth is about 0.95^, or of the diameter. 

By the help of Table 115 the velocity and discharge may be com¬ 
puted when c is known, but it is not possible on account of the lack 
of experimental knowledge to state precise values of c for different 
values of r in circles of different sizes. However, it is known that an 
increase in r increases c, and that a decrease in r decreases c. The 
following experiments of Darcy and Bazin show the extent of this 
variation for a semicircular conduit of 4.1 feet diameter, and they also 
teach that the nature of the interior surface greatly influences the values 
of c. Two conduits were built, each with a slope 5 = 0.0015 and d = 
4.1 feet. One was lined with neat cement, and the other with a mor¬ 
tar made of cement with one-third fine sand. The flow was allowed 
to occur with different depths, and the discharges per second were 
gaged by means of orifices; this enabled the velocities to be computed, 
and from these the values of the coefficient c were found. The fol¬ 
lowing are a portion of the results obtained, d' denoting the depth 
of water in the conduit, r the hydraulic radius, v the mean velocity, 
and all linear demensions being in English feet: 


For cement lining For mortar lining 


d' 

r 

V 

c 

&* 

r 

V 

c 

2.05 

1.029 

6.06 

154 

2.04 

1.022 

5-55 

142 

1.61 

0.867 

5-29 

147 

1.69 

0.900 

4.94 

i 35 

1.03 

0.605 

4.16 

138 

1.00 

0.635 

3-87 

125 

0.59 

0.366 

3.02 

129 

0.61 

0-379 

2.87 

120 


It is here seen that c decreases quite uniformly with r, and that the 
velocities for the mortar lining are 8 or 10 per cent less than those for 
the neat cement lining. 

The value of the coefficient c for these experiments may be roughly 
expressed for English measures by 

C = Ci — 16 (| d — d') 

in which c L is the coefficient for the conduit when running half fulL 
How this will apply to different diameters and velocities is not known; 
when d' is greater than oM, it will probably prove incorrect. In 
practice, however, computations on the flow in partly filled conduits 
are of rare occurrence. 

Prob. 115 . Compute the hydraulic radius for a circular conduit of 4.1 
feet diameter, when it is three-fourths filled with water, and also the mean 


282 Chap. 9 . Flow in Conduits and Canals 

velocity when it is lined with neat cement and laid on a grade of 0.15 feet 
per 100 feet. 


Art. 116 . Rectangular Conduits 


In designing an open rectangular trough or conduit to carry 
water there is a certain ratio of breadth to depth which is most 
advantageous, because thereby either the discharge is the greatest 
or the least amount of material is required for its construction. 
Let b be the breadth and d the depth of the water section, then the 
area a is bd and the wetted perimeter p is b + 2d. If the area a is 
given, it may be required to find the relation between b and d 
so that the discharge may be a maximum. If the wetted perim¬ 
eter p is given, the relation between b and d to produce the same 
result may be demanded. It is now to be shown that in both 
cases the breadth is double the depth, or b — 2d. This is called 
the most advantageous proportion for an open rectangular con¬ 
duit, since there is the least head lost in friction when the velocity 
and discharge are the greatest possible. 

Let r be the hydraulic radius of the cross-section, or 

_ a _ bd 
p b + 2 d 


then, from the Chezy formula ( 113 ), the expressions for the veloc¬ 
ity and discharge are 


v = 



bd 

b T 2d 



b 3 d 3 
b -b 2d 


In these expressions it is required to find the relation between 
b and d, which renders both v and q a maximum. 


Let the wetted perimeter p be given, as might be the case 
when a definite amount of lumber is assigned for the construction 
of a trough; then b + 2d = p, or d = \{p - b), and 



= c Vs 


\ 


b 3 (p — b) 3 
8 p 


in which p is a constant. Differentiating either of these expres¬ 
sions with respect to b and equating the derivative to zero, there 












Rectangular Conduits. Art. 116 


283 


is found b = \p, and hence d = \p. Accordingly b = 2d, or the 
breadth is double the depth. 


Again, let the area a be given, as might be the case when a 
definite amount of rock excavation is to be made; then bd = a, 
or d = a/b, and 


v 


= c Vs 


ab 


\b 2 + 


= cVs 


a 3 b 


2 a 


b 2 -{- 2a 


in which a is constant. By equating the first derivative to zero, 
there is found b 2 = 2a, and hence d 2 = \a. Accordingly b = 2d , 
or the breadth is double the depth, as before. 

It is seen in the above cases that the maximum of both v and q 
occur when r is a maximum, or when r = \d. It is indeed a 
general rule that r should be a maximum in order to secure the 
least loss of head in friction. The circle has a greater hydraulic 
radius than any other figure of equal area. 


In these investigations c has been regarded as constant, al¬ 
though strictly it varies somewhat for different ratios of b to d. 
The rule deduced is, however, sufficiently close for all practical 
purposes. It frequently happens that it is not desirable to adopt 
the relation b = 2d , either because the water pressure on the sides 
of the conduit becomes too great or because it is advisable to 
limit the velocity so as to avoid scouring the bed of the channel. 
Whenever these considerations are more important than that 
of securing the greatest discharge, the depth is made less than one- 
half the breadth. 

The velocity and discharge through a rectangular conduit 
are expressed by the general equations 

v = c Vrs q — av = ca^/rs 

and are computed without difficulty for any given case when the 
coefficient c is known. To determine this, however, is not easy, 
for it is only from recorded experiments that its value can be 
ascertained. When the depth of the water in the conduit is one- 
half of its width, thus giving the most advantageous section, the 
values of c for smooth interior surfaces may be estimated by the 
use of Table 114 for circular conduits, although c is probably 






284 Chap. 9 . Flow in Conduits and Canals 

smaller for rectangles than for circles of equal area. When the 
depth of the water is less or greater than \d, it must be remem¬ 
bered that c increases with r. The value of c also is subject to 
slight variations with the slope s, and to great variations with the 
degree of roughness of the surface. 

Table 116 , derived from Smith’s discussion of the experiments 
of Darcy and Bazin, gives values of c for a number of wooden 
and masonry conduits of rectangular sections, all of which were 
laid on the grade of 0.49 per cent or s = 0.0049. The great influence 

Table 116 . Coefficients c for Rectangular Conduits 


Enplaned Plank 
b = 3-93 Feet 

Unplaned Plank 
b = 6.53 Feet 

Neat Cement 
b — 5.94 Feet 

Brick 

b = 6.27 Feet 

, ! 

a 1 c 

1 

d 

c 

d 

c 

d 

c 

O.27 99 

0.20 

89 

O.18 

116 

0.20 

89 

-L 

M 

O 

00 

•30 

IOI 

.28 

125 

•31 

98 

.67 ! 112 

.46 

109 

•43 

132 

•49 

104 

.89 ! 114 

.60 

II3 

•56 

135 

•57 

105 

1.00 114 

.72 

116 

•63 

136 

.65 

105 

1.19 116 

.78 

116 

.69 

136 

•7i 

106 

1.29 117 

.89 

118 

.80 

137 

.85 

107 

1.46 1 118 

•94 

120 

.91 

138 

•97 

no 


of roughness of surface in diminishing the coefficient is here 
plainly seen. For masonry conduits with hammer-dressed sur¬ 
faces c may be as low as 60 or 50, particularly when covered 
with moss and slime. 

Prob. 116 . Find the size of a trough, whose width is double its depth, 
which will deliver 125 cubic feet per minute when its slope is 0.002, taking 
the coefficient c as 100. 


Art. 117 . Trapezoidal Sections 

Ditches and conduits are often built with a bottom nearly 
flat and with side slopes, thus forming a trapezoidal section. 
The side slope is fixed by the nature of the soil or by other cir¬ 
cumstances, the grade is given, and it may be then required to 























Trapezoidal Sections. Art. 117 


285 


ascertain the relation between the bottom width and the depth 
of water, in order that the section shall be the most advantageous. 
This can be done by the same reasoning as used for the rectangle 
in the last article, but it may be well to employ a different method, 
and thus be able to consider the subject in a new light. 

Let the trapezoidal channel have the bottom width b, the 
depth d, and let 6 be the angle made by the side slopes with the 
horizontal. Let it be required to 
discharge q cubic units of water per 
second. Now q = caVrs, and the 
most advantageous proportions may 
be said to be those that will render 
the cross-section a a minimum for a given discharge, for thus the 
least excavation, will be required. From Fig. 117 , 

a = d (bd cotO) p = b-\-2d/sinfl 

and from these the value of r may be expressed in terms of a, 
d, and 6 ; inserting this in the formula for q, it reduces to 



c 2 sa 3 _ qhi _ J 2 
d d 2 ^ Vsin# 



in which the second member is a constant. Obtaining the first 
derivative of a with respect to d, and then replacing q 2 by its 
value c 2 a 2 rs, there results 

d = 2 q 2 /c 2 a 2 s d—2r 

which is the relation that renders the area a a minimum; that is, 
the advantageous depth is double the hydraulic radius. Now 
since a/p = r, it is easy to show that 

b + 2 d cot# = 2^/sin# 

or, the top width of the water surface should equal the sum of the 
two side slopes in order to give the most advantageous section. 
Since c has been regarded constant, the conclusion is not a rigor¬ 
ous one, although it may safely be followed in practice. As 
in all cases of an algebraic minimum, a considerable variation 
in the value of the ratio d/b may occur without materially effect¬ 
ing the value of the area a. In many cases it is not possible to 














286 


Chap. 9. Flow in Conduits and Canals 


have so great a depth of water as the rule d = 2 r requires because 
of the greater cost of excavation at such depth, or because width 
rather than depth may be needed for other reasons. 


When a trapezoidal channel is to be built, the general formulas 
v — c y/rs and q = av may be used to obtain a rough approximation 
to the discharge, c being assumed from the best knowledge at hand. 
The formula of Kutter (Art. 118) or that of Bazin (Art. 122) may be 
used to determine c when the nature of the bed of the channel is known. 
For a channel already built, computations cannot be trusted to give 
reliable values of the discharge on account of the uncertainty re¬ 
garding the coefficient, and in an important case an actual gaging 
of the flow should be made. This is best effected by a weir, but if 
that should prove too expensive, the methods explained in the next 
chapter may be employed to give more precise results than can usually 
be determined by computation from any formula. 


The problem of determining the size of a trapezoidal channel 
to carry a given quantity of water does not require c to be de¬ 
termined with great precision, since an allowance should be made 
on the side of safety. For this purpose the following values may 
be used, the lower ones being for small cross-sections with rough 
and foul surfaces, and the higher ones for large cross-sections 
with quite smooth and clean earth surfaces: 


For unplaned plank, c = 

For smooth masonry, c = 

For clean earth, c = 

For stony earth, c = 

For rough stone, c = 


For earth foul with weeds, c = 


ioo to 120 
90 to no 
60 to 80 
40 to 60 
35 to 50 
30 to 50 


To solve this problem, let a and p be replaced by their values 
in terms of b and d. The discharge then is 


q = cd{b-\-d cot#) 


\d (b-\-d cot#) x sin# 

b sin# + 2 d 


Now when q, c, #, and s are known, the equation contains two 
unknown quantities, b and d. If the section is to be the most 
advantageous, b can be replaced by its value in terms of d as 
above found, and the equation then has but one unknown. 





Kutter’s Formula. Art. 118 


287 


Or in general, if b — md, where m is any assumed number, a solu¬ 
tion for the depth gives the formula 

= <f (msinfl+ 2) 

C 2 s ( m + cot#) 3 sin# 


For the particular case where the side slopes are 1 on 1 or # = 45°, 
and the bottom width is to be equal to the water depth, or m — 1, 
this becomes d = ag6 ((f/ch) \ 


These formulas are analogous to those for finding the diameter 
of pipes and circular conduits, and the numerical operations are 
in all respects similar. It is plain that by assigning different 
values to m numerous sections may be determined which will 
satisfy the imposed conditions, and usually the one is to be se¬ 
lected that will give both a safe velocity and a minimum cost. 
In Art. 120 will be found an example of the determination of the 
size of a trapezoidal canal. 

Prob. 117 . If the value of c is 71, compute the depth of a trapezoidal 
section to carry 200 cubic feet of water per second, 0 being 45 0 , the slope 
^ being 0.001, and the bottom width being equal to the depth. Compute also 
the area of the cross-section and the mean velocity. 


Art. 118 . Kutter’s Formula 

An elaborate discussion of all recorded gagings of channels 
was made by Ganguillet and Kutter in 1869, from which an im¬ 
portant empirical formula was deduced for the coefficient c 
in the Chezy formula v = cVrs. The value of c is expressed in 
terms of the hydraulic radius r, the slope s, and the degree of 
roughness of the surface, and may be computed when these three 
quantities are given. When r is in feet and v in feet per second, 
Kutter’s formula for the Chezy coefficient c is 


1.811 , * , 0.00281 

+ 41.65 H- 


n 


c = 


1 + 


n 


Vr 


( 


41.65+ 


0.00281 


( 118 ) 


in which n is an abstract number whose value depends only 
upon the roughness of the surface. By inserting this value of 








288 Chap. 9 . Flow in Conduits and Canals 

c in the Chezy formula for v, the mean velocity is made to de¬ 
pend upon r, s, and the roughness of the surface. The following 
values of n were assigned by Kutter to different surfaces: 

n = o.ooq for well-planed timber, 
n = o.oio for neat cement, 
n = o.oii for cement with one-third sand, 
n = 0.012 for unplaned timber, 

7i = 0.013 for ashlar and brick work, 
n = 0.015 for unclean surfaces in sewers and conduits, 

7 i = 0.017 f° r rubble masonry, 

tv = 0.020 for canals in very firm gravel, 

71 = 0.025 i° r canals and rivers free from stones and weeds, 

71 = 0.030 for canals and rivers with some stones and weeds, 
n = 0.035 f° r canals and rivers in bad order. 

The formula of Kutter has received a wide acceptance on 
account of its application to all kinds of surfaces. Notwith¬ 
standing that it is purely empirical, and hence not perfect, it is 
to be regarded as a formula of great value, so that no design for 
a conduit or channel should be completed without employing 
it in the investigation, even if the final construction be not based 
upon it. In sewer work it is extensively employed, n being taken 
as about 0.015. The formula shows that the coefficient c al¬ 
ways increases with r, that it decreases with 5 when r is greater 
than 3.28 feet, and that it increases with 5 when r is less than 3.28 
feet. When r equals 3.28 feet, the value of c is simply 1.811 /n. 
It is not likely that future investigations will confirm these laws 
of variation in all respects. 

In the following articles are given values of c for a few cases, 
and these might be greatly extended, as has been done by Kutter 
and others.* But this is scarcely necessary except for special 
lines of investigation, since for single cases there is no difficulty 
in directly computing it for given data. For instance, take a 
rectangular trough of unplaned plank 3.93 feet wide on a slope 
of 4.9 feet in 1000 feet, the water being 1.29 feet deep. Here 

* Flow of Water in Rivers and Other Channels. Translated, with 
additions, by Hering and Trautwine, New York, 1889. 


Sewers. Art. 119 


289 


5 — 0.0049, an d f —0.779 feet. Then n being 0.012, the value of 
c to be used in the Chezy formula is found to be 


c = 


1.811 , , 1 0.00281 

+ 41.65 +- 

0.0049 


0.012 


0.012 ( £ , 0.00281 

— 7=1 4165 +- 

Vo.779 v 0.0049 > 


= I2 3 


The data here used are taken from Table 116 , where the actual 
value of c is given as 117; hence in this case Kutter’s formula 
is about 5 per cent in excess. As a second example, the follow¬ 
ing data from the same table will be taken: a rectangular con¬ 
duit in neat cement, 6 = 5.94 feet, ^ = 0.91 feet, 5 = 0.0049. Here 
n = 0.010, and r = 0.697 feet. Inserting all values in the formula, 
there is found 0 = 148, which is 8 percent greater than the true 
value 138. Thus is shown the fact that errors of 5 and 10 per¬ 
cent are to be regarded as common in calculations on the flow 
of water in conduits and canals. 


Prob. 118 . The Sudbury conduit is of horse-shoe form and lined with 
brick laid with cement joints one-quarter of an inch thick, and laid on a 
slope of 0.0001895. Compute the discharge in 24 hours when the area is 33.31 
square feet and the wetted perimeter 15.21 feet. 


Art. 119 . Sewers 

Sewers smaller in diameter than 18 inches are always circular 
in section. When larger than this, they are built with the sec¬ 
tion either circular, egg-shaped, or of the horse-shoe form. The 
last shape is very disadvantageous when a small quantity of 
sewage is flowing, for the wetted perimeter is then large compared 
with the area, the hydraulic radius is small, and the velocity 
becomes low, so that a deposit of the foul materials results. As 
the slope of sewer lines is often very slight, it is important that 
such a form of cross-section should be adopted to render the veloc¬ 
ity of flow sufficient to prevent this deposit. A velocity of 2 
feet per second is found to be about the minimum allowable 
limit, and 4 feet per second need not be usually exceeded. 

The egg-shaped section is designed so that the hydraulic 
radius may not become small even when a small amount of 









290 


Chap. 9 . Flow in Conduits and Canals 


sewage is flowing. One of the most common forms is that shown 
in Fig. 119 , where the greatest width DD is two-thirds of the depth 

EM. The arch DHD is a semicircle 
described from A as a center. The 
invert LML is a portion of a circle 
described from B as a center, the 
distance BA being three-fourths of 
DD and the radius BM being one- 
half of AD. Each side DL is de¬ 
scribed from a center C so as to be 
tangent to the arch and invert. 
These relations may be expressed 
more concisely by 



EM = i ID 


AB = ID 


BM = \D 


CL = ihD 


in which D is the horizontal diameter DD. 


Computations on egg-shaped sewers are usually confined to 
three cases, namely, when flowing full, two-thirds full, and one- 
third full. The values of the sectional areas, wetted perimeters, 
and hydraulic radii for these cases, as given by Flynn,* are 


a 

Full 1.1485 D- 

Two-thirds full 0.7558 D- 

One-third full 0.2840 Z) 2 


P r 

3.965 D 0.2897 D 

2-394 D 0.3157 D 

1.375 D 0.2066 D 


This shows that the hydraulic radius, and hence the velocity, 
is but little less when flowing one-third full than when flowing 
with full section. 

Egg-shaped sewers and small circular ones are formed by 
laying consecutive lengths of clay or cement pipe whose interior 
surfaces are quite smooth when new, but may become foul after 
use. Large sewers of circular section are made of brick, and are 
more apt to become foul than smaller ones. In the separate 
system, where systematic flushing is employed and the pipes are 
small, foulness of surface is not so common as in the combined 
system, where the storm water is alone used for this purpose. 


* Van Nostrand’s Magazine, 1883, vol. 28, p. 138. 





Sewers. Art. 119 


291 


In the latter case the sizes are computed for the volume of 
storm water to be discharged, the amount of sewage being very 
small in comparison. 

The discharge of a sewer pipe enters it at intervals along 
its length, and hence the flow is not uniform. The depth of 
the flow increases along the length, and at junctions the size 
of the pipe is enlarged. The strict investigation of the problem 
of flow is accordingly one of great complexity. But considering 
the fact that the sewer is rarely filled, and that it should be made 
large enough to provide for contingencies and future extensions, 
it appears that great precision is unnecessary. The practice, 
therefore, is to discuss a sewer for the condition of maximum 
discharge, regarding it as a channel with uniform flow. The 
main problem is that of the determination of size; if the form is 
circular, the diameter is found, as in Art. 114 , by 

d = (8 q/irc Vs)* = 1.45 (q/c Vs ) 5 

If the form is egg-shaped and of the proportions above ex¬ 
plained, the discharge when running full is 

q = ac Vrs = 1.1485 D 2 cV0.2897 
from which the value of D is found to be 

D = i.2i(<7/c Vs) z 

Thus, when q has been determined and c is known, the required 
sizes for given slopes can be computed. The velocity should also 
be found in order to ascertain if it is low enough to prevent 
scouring (Art. 135 ). 

Experiments from which to directly determine the coefficient c 
for the flow in sewers are few in number, but since the sewage is 
mostly w r ater, it may be approximately ascertained from the values 
for similar surfaces. Kutter’s formula has been extensively employed 
for this purpose, using 0.015 for the coefficient of roughness. Table 
119 gives values of c for three different slopes and for two classes of 
surfaces. The values for the degree of roughness represented by n = 
0.017 are applicable to sewers with quite rough surfaces of masonry; 
those for n = 0.015 are applicable to sewers with ordinary smooth 
surfaces, somewhat fouled or tuberculated by deposits, and are the 



292 


Chap. 9 . Flow in Conduits and Canals 


Table 119 . Kutter’s Coefficients c for Sewers 


Hydraulic 
Radius r 
in Feet 

j = 0.00005 

s = o.ooox 

5 = 

0.01 

n — 0.015 

n - 0.017 

. 

n = 0.015 

n — 0.0x7 

n = 0.015 

n = 0.017 

0.2 

52 

43 

58 

48 

68 

57 

0-3 

60 

5 i 

66 

56 

76 

64 

0.4 

65 

56 

73 

61 

83 

70 

0.6 

76 

65 

82 

70 

90 

76 

0.8 

82 

72 

87 

76 

95 

82 

1. 

88 

77 

92 

80 

99 

00 

i -5 

100 

86 

103 

89 

108 

93 

2. 

106 

94 

108 

96 

hi 

99 

3 - 

116 

103 

118 

104 

118 

105 


ones to be generally used in computations. By the help of this table 
and the general equations for mean velocity and discharge, all prob¬ 
lems relating to flow in sewers can be readily solved. 

Prob. 119 . The grade of a sewer is i foot in 1004, and its discharge 
is to be 130 cubic feet per second. What should be the diameter of the 
sewer if it is circular ? 

Art. 120 . Ditches and Canals 

Ditches for irrigating purposes are of a trapezoidal section, 
and the slope is determined by the fall between the point from 
which the water is taken and the place of delivery. If the fall 
is large, it may not be possible to construct the ditch in a straight 
line between the two points, even if the topography of the country 
should permit, on account of the high velocity which would re¬ 
sult. A velocity exceeding 2 feet per second may often injure 
the bed of the channel by scouring, unless it be protected by 
riprap or other lining. For this reason, as well as for others, the 
alignment of ditches and canals is often circuitous. 

The principles of the preceding articles are sufficient to 

solve all usual problems of uniform flow in such channels when the 

% 

values of the Chezy coefficient c are known. These are perhaps 
best determined by Kutter s formula, and for greater convenience 
Table 120 has been prepared which gives their values for three 
























Ditches and Canals. Art. 120 


293 


Table 120 . Kutter’s Coeeficients c for Channels 


Hydraulic 
Radius r 
in Feet 

5 = 0.00005 

j = 0.0001 

5 = 

O.OI 

n = 0.025 

n = 0.030 

n = 0.025 

n = 0.030 

n - 0.025 

n = 0.030 

o -5 

38 

3 i 

41 

33 

47 

37 

1. 

49 

40 

52 

42 

56 

45 

i -5 

57 

47 

59 

48 

62 

5 i 

2. 

64 

52 

65 

53 

67 

54 

3 - 

72 

59 

72 

59 

72 

60 

4 . 

77 

64 

77 

64 

76 

63 

5 - 

81 

68 

80 

68 

79 

66 

6. 

86 

72 

84 

7 i 

80 

68 

8. 

9 i 

76 

00 

74 

82 

70 

10. 

96 

80 

91 

80 

85 

73 

15. 

105 

89 

97 

84 

90 

77 

25 - 

114 

100 

IOI 

92 

95 

82 


slopes and two degrees of roughness. By interpolation in this 
table values for intermediate data may also be found; for instance, 
if the hydraulic radius be 3.5 feet, the slope be 1 on 1000, and 
n be 0.025, the value of c is found to be 74.5. 

As an example of the use of the table let it be required to find 
the width and depth of a ditch of most advantageous cross- 
section, whose channel is to be in tolerably good order, so that 
n — 0.025. The amount of water to be delivered is 200 cubic 
feet per second and the grade is 1 in 1000, the side slopes of the 
channel being 1 on 1. From Art. 117 the relation between the 
bottom width and the depth of the water is, since # is 45 0 , 

b = d(—r— — 2 cot#) = 0.82 8d 
Vsm# J 


The area of the cross-section then is 

a = d{b + d cot#) = 1.82 Sd 2 
and the wetted perimeter of the cross-section is 

p = b-i —\ d — = 3.656 d 
sin# 























294 


Chap. 9 . Flow in Conduits and Canals 


whence the hydraulic radius is 0.5 d, as must be the case for all 
trapezoidal channels of most advantageous section. Now, 
since d is unknown, c cannot be taken from the table, and as a 
first approximation let it be supposed to be 60. Then in the gen¬ 
eral formula for q the above values are substituted, giving 

200 = 60 X 1.82 Sd 2 Vo.5 d X c.oo 1 

from which d is found to be 5.8 feet. Accordingly r— 2.9 feet, 
and from the table c is about 71. Repeating the computation 
with this value of c, there is found d = 5.44 feet, which, considering 
the uncertainty of c, is sufficiently close. The depth may then 
be made 5.5 feet, the bottom width is 

b = 0.828 X 5.5 = 4.55 feet, 

and the area of the cross-section is 

a = 1.828 X 5.5 s = 55.3 square feet, 

which gives for the mean velocity 

v = = 3.62 feet per second. 

55-3 

This completes the investigation if the velocity is regarded 
as satisfactory. But for most earths this would be too high, 
and accordingly the cross-section of the ditch must be made 
wider and of less depth in order to make the hydraulic radius 
smaller and thus diminish the velocity. 

The following statements show approximately the velocities 
which are required to move different materials: 

0.25 feet per second moves fine clay, 

0.5 feet per second moves loam and earth, 

1.0 feet per second moves sand, 

2.0 feet per second moves gravel, 

3.0 feet per second moves pebbles 1 inch in size, 

4.0 feet per second moves spalls and stones, 

6.0 feet per second moves large stones. 

The mean velocity in a channel may be somewhat larger than 
these values before the materials will move, because the velocities 
along the wetted perimeter are smaller than the mean velocity. 
More will be found on this subject in Art. 135 . 




Large Steel, Wood, and Cast-iron Pipes. Art. 121 295 


Prob. 120 . A ditch is to discharge 200 cubic feet per second with a 
mean velocity of 3.4 feet per second. If its bottom width is 16 feet and the 
side slopes are 1 on 1, compute the depth of water and the slope of the ditch. 

Art. 121 . Large Steel, Wood, and Cast-iron Pipes • 

Long pipes of large size are usually regarded as conduits even 
when running under pressure, for in formula (97) 2 the ratio h/l 
may be replaced by the slope s and the diameter d is four times 
the hydraulic radius r ; then it becomes 

v = V8g// Vrs = c Vrs 

which is the same as the Chezy formula. Values of c may be 
directly computed from observed values of v, r, and s, and this 
has been done by many experimenters. When values of c are 
known, all computations for long pipes may be made exactly 
like those for circular conduits. 

In the following Table 121 a * are shown the results of experi¬ 
ments on a number of steel pipes ranging from 33 to 108 
inches in diameter and from new to 15 years of age. The 
experiments were made at velocities ranging from 1.0 to 6.0 feet 
per second, and the values given in the table are those read from 
mean curves of the plottings of the results of the experiments. 
In the column headed “Material and Joint” the letters S and W 
refer to steel and wrought iron respectively, while the letters 
B, C, and T refer to the style of the joint used in the construction 
of the pipe, B indicating butt, C cylinder, and T taper joint, 
respectively. The experiments bracketed together in the first 

* Following are the sources from which the results tabulated in this 
table have been obtained : 

Nos. 1, 2, and 10. Transactions American Society of Civil Engineers, 

Vol. 26, p. 203. 

Nos. 3, 4, 5, 6, 7, 8, 11, 12, 13, 15, 16, 24. Herschel’s 115 Experiments, 

New York, 1897. 

Nos. 9, 14, and 17 are here shown through courtesy of Morris R. Sher- 
rerd, Chief Engineer, Department of Public Works, Newark, N.J. 

Nos. 18, 19. Transactions American Society of Civil Engineers, Vol. 

40, p. 471, and Vol. 44, p. 34. 

Nos. 20, 21, 22, and 23. By courtesy of The Jersey City Water Supply 
Co., Paterson, N.J. 



Table 121 a. Actual Coefficients c for Riveted Steel Pipes 


296 


Chap. 9 . Flow in Conduits and Canals 



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OJ 

<u 

CJ 

CTj 

C/I 

C/I 





£ 

& 

55 

55 

£ 

52J 

55 


55 

£ 

55 

55 

& 

55 

55 



55 

55 

55 


cG 

*—< 


























Jh 

G 

• f» 


























Vh 

t/l 

























G 

<D 

CO 

'O 

NO 

NO 

NO 

00 

Cl 

Cl 

Cl 

Cl 

00 

OO 

00 

00 

OO 

00 

00 

Cl 

Cl 

Cl 

Cl 

Cl 

Cl 

00 


CJ 

to 

to 

to 

CO 

CO 

CO 



'G- 


•T 

H* 










X^ 

X^ 

0 

G 

G 
























HH 

c3 

-H 

























Q 


























O 

u 

Uj 

























G g 
<L> -5 

»- G 

HH 

0* 

to 


VO 

NO 

x^. 

00 

O' 

0 

HH 

Cl 

CO 


VO 

NO 


00 

O' 

0 

M 

Cl 

co 


0) 

<D 


























G 

























5 


























































Large Steel, Wood, and Cast-iron Pipes. Art. 121 297 


column were made at different ages as shown on the same pipe 
and indicate the deterioration which is to be expected with age. 
(See Art. 107 .) Experiments numbered 12 and 15 are one and 
the same and are shown twice in order that comparison may 
more readily be made with experiments 13 and 14 and 16 and 17. 
Experiments 12 and 15 were made on the entire length of the 
pipe referred to, while 13 and 14 were made on its upper end and 
16 and 17 on its lower end. 

As illustrating the values of n in Kutter’s formula for some 
of the experiments shown in Table 121 a the following, for experi¬ 
ments 18 and 19, are here given: 

Velocity in feet per second = 1.0 2.0 3.0 4.0 5.0 

Exp. 18, n = 0.013 0.014 0.015 0.014 

Exp. 19, n = 0.018 0.016 0.015 0.015 0.015 

For wooden stave pipes the gagings of Noble and those of 

Marx, Wing, and Hoskins, already referred to in Art. 108 , furnish 
the following values of the coefficient c, those given for the 6-foot 
diameter in the first line being for new pipe and those in the 
second line after two years’ use. 

Velocity in feet per second, v = 1 2 3 4 5 


3.7 feet diameter c = 


(109) 

113 

4.5 feet diameter c = (112) 

122 

126 

128 

6.0 feet diameter c = 100 

115 

122 

125 

6.0 feet diameter c = 116 

120 

121 

122 


Here the two values in parentheses have been found by a graphic 
discussion of the results of the observations. For the first of 
these pipes the valve of Kutter’s n ranges from 0.013 1 ° 0.012, 
while for the second and third it is practically constant at 0.013. 

Many gagings have been made on cast-iron pipes, and the re¬ 
sults show great variations which can be ascribed to many causes ; 
among these may be mentioned the progressive deterioration 
due to age as well as that due to the particular kind of water 
carried by the pipe, the care with which the pipe has been laid, 
and with which the joints have been made. In Table 1216 are 
shown the values of the coefficient c for certain pipes of different 
diameters and ages and for varying velocities. The friction 
factors for these same gagings are given in Art. 106 . 


298 


Chap. 9 . Flow in Conduits and Canals 


Table 1216 . Actual Coefficients c for Cast-iron Pipes 


Diameter 

in 

Inches 

Age 

in 

Years 

Velocity in Feet per Second 

Reference 

1.0 

2.0 

30 

4.0 

12 

O 

IOI 

no 

IIS 

118 

Trans. Am. Soc. C.E., vol. 47 

12 

15 

65 

58 



Hering’s Kutter * 

12 

22 

49 

46 

45 


Hering’s Kutter * 

20 

5 


ns 

109 


Trans. Am. Soc. C.E., vol. 35 

20 

25 


61 

60 

59 

Hering’s Kutter * 

36 

4 




130 

Trans. Am. Soc. C.E., vol. 44 

36 

3 s 




66 

Trans. Am. Soc. C.E., vol. 44 

48 

0 




141 

Trans. Am. Soc. C.E., vol. 35 

48 

7 


96 



Trans. Am. Soc. C.E., vol. 28 

48 

16 


107 

105 

105 

Trans. Am. Soc. C.E., vol. 35 


Prob. 121 . Compare the diameter of a cylinder joint riveted steel pipe 
25 000 feet long to carry 30 000 000 gallons daily at a loss of head of 5 feet 
per mile with the diameter of a cast-iron pipe for the same service. 

Art. 122 . Bazin’s Formula 

In 1897 Bazin proposed a formula for open channels as the 
result of an extended discussion of the most reliable gagings.f 
In it the coefficient c is expressed in terms of the hydraulic radius 
and the roughness of the surface, but the slope does not enter: 

v = c Vrs c =-—-- (122) 

0.552 + ml a Jr 

This is for English measures, r being in feet and v in feet per sec¬ 
ond, and the quantity m has the following values: 

m = 0.06 for smooth cement or matched boards, 
m = 0.16 for planks and bricks, 
m = 0.46 for masonry, 
m = 0.85 for regular earth beds, 
m = 1.30 for canals in good order, 
m = 1.75 for canals in very bad order, 

* Hering and Trautwine’s translation of Ganguillet and Kutter’s Flow of 
Water in Rivers and Other Channels, New York, 1889, p. 155. 
f Annales des ponts et chaussees, 1897, 4® trimestre, pp. 20-70. 


















Bazin’s Formula. Art. 122 


299 


Table 122 gives values of c computed from ( 122 ) for these values 
of m and for several values of r, from which coefficients may be 
selected for particular surfaces. It may be noted that for a per- 


Table 122. Bazin’s Coefficients c for Channels 


Hydraulic 
Radius r 
in Feet 

m = 0.06 

m = 0.16 

tn — 0.46 

vo 

00 

6 

II 

£ 

m = 1.30 

m = 1.75 

0-5 

136 

III 

72 




I. 

142 

122 

86 

62 



i -5 

145 

127 

94 

70 

54 


2. 

146 

131 

100 

76 

60 

49 

3 - 

148 

135 

107 

84 

67 

56 

4 - 

149 

137 

hi 

89 

72 

61 

5 - 

150 

I40 


94 

78 

67 

6. 

151 

141 

117 

96 

80 

69 

8. 

!52 

143 

122 

101 

85 

73 

10. 

152 

I44 

125 

106 

9 i 

79 

i 5 - 



131 

113 

98 

87 

25 - 




121 

107 

97 


fectly smooth surface where m = o, the formula gives v = 158 'Vrs, 
which cannot be correct since uniform velocity could not obtain 
on such a surface. For this extreme case Kutter’s formula ap¬ 
pears to be more satisfactory, for if n = o the value of c is in¬ 
finite. However, no empirical formula can be tested by applying 
it to an extreme case. 

A comparison of the values of c obtained from the formulas 

of Kutter and Bazin only serves to emphasize the uncertainty 

regarding the selection of the proper coefficient in particular 

cases. Kutter’s n = 0.010 corresponds to Bazin’s m = 0.06, 

and for several different hydraulic radii the coefficients for this 

degree of roughness are as follows: 

Hydraulic radius r in feet, 1 3 5 7 

From Bazin’s formula, c = 142 148 150 151 

From Kutter, s = 0.01, 0=156 179 187 191 

From Kutter, s = 0.001, 0=155 178 187 192 

From Kutter, ^ = 0.00005, c = 140 178 193 203 

While the agreement is fair for a hydraulic radius of one foot, it 
fails to be satisfactory for larger radii. This is perhaps a severe 

















300 


Chap. 9 . Flow in Conduits and Canals 


^comparison because it is probable that no channel in neat cement 
has ever been constructed having a hydraulic radius as great as 
7 feet, but it serves to show that these empirical formulas differ 
widely when applied to unusual cases. For the present, at least, 
the formula of Kutter appears to receive the most general accept¬ 
ance, but undoubtedly the time will come when it will be re¬ 
placed by a more satisfactory one. An actual gaging of the dis¬ 
charge by the method of Art. 131 will always give more reliable 
information than can be obtained from any formula. 

For a hydraulic radius of 3.28 feet Kutters formula for c 
reduces to the convenient expression 


c = 1.81 i/n 


whence 


v = 


1.811 

n 



and this may be used for approximate computations when r lies 
between 2 and 6 feet. Here n is the roughness factor, the values 
of which are given in Art. 118 . When r — 3.28 feet, Bazin’s 
formula gives c = 136 for brickwork, while Kutter’s gives c = 140; 
for canals in good order Bazin’s formula gives c = 69, while 
Kutter’s gives 0 = 72. The comparison is very satisfactory, and 
so close an agreement is not generally to be expected when com¬ 
putations are made from different formulas. The formula of 
Bazin is largely used in France and England, and that of Kutter 
in other countries.. 

Prob. 122 . Solve Problem 118 by the use of Bazin’s coefficients. 


Art. 123 . Masonry Conduits 

Masonry conduits or aqueducts for conveying water have 
been used since the days of ancient Rome. In cases where large 
quantities of water are to be carried on small slopes and where 
the topography of the country is at a suitable elevation they 
offer the most economical means for its conveyance. The Sud¬ 
bury and Wachusett aqueducts for the supply of Boston, the 
Jersey City aqueduct for the supply of that city, the old Croton 
and the New Croton aqueducts for the supply of New York City 
are among the largest and longest which have yet been constructed. 



Masonry Conduits. Art. 123 


301 


In 1912 there are being built the Catskill aqueduct also for New 
York City and the Los Angeles aqueduct for the city of Los 
Angeles in California. Large portions of these aqueducts are in 
tunnels on the hydraulic gradient, and in the case of the Catskill 
aqueduct of a total of no miles of main conduit nearly 30 per¬ 
cent is in rock tunnel from 300 to 1100 feet below the surface. 
These tunnels are circular in cross-section, and their diameters 
range from 11 to 15 feet. 

Relatively few experiments for determining the coefficients 
of flow have been made on these aqueducts. From their gagings 
of the Sudbury aqueduct, Fteley and Stearns * determined a 
formula for mean velocity. The cross-section of this aqueduct, 
which is laid on a slope of 0.0002, consists of a part of a circle 9.0 
feet in diameter, having an invert of 13.22 feet radius, whose 
span is 8.3 feet and depression 0.7 feet, the axial 
depth of the conduit being 7.7 feet. It is lined 
with brick, having cement joints \ of an inch 
thick. The flow was allowed to occur with 
different depths, for each of which the discharge 
was determined by weir measurement. A dis¬ 
cussion of the results led to the conclusion that in the portion 
with the brick lining the coefficient c had the value ^r 0,12 when 
r is in feet, and hence results the exponential formula 

v = 127 r 0,12 -\/rs =127 f°- 62 s 0 - 50 

In a portion of this conduit where the brick lining was coated with 
pure cement, the coefficient was found to be from 7 to 8 percent 
greater than i2 7r 0 - 12 . In another portion where the brick lining 
was covered with a cement wash laid on with a brush, the co¬ 
efficient was from 1 to 3 percent greater. For a long tunnel in 
which the rock sides were ragged, but with a smooth cement in¬ 
vert it was found to be about 40 percent less. 

Gagings on the New Croton Aqueduct f showed that the mean 
velocity when the aqueduct was new could be represented by the 



* Transactions American Society of Civil Engineers, 1883, vol. 13, p. 114. 
f Engineering Record, 1895, vol. 32, p. 223. 





302 Chap. 9 . Flow in Conduits and Canals 

expression v = i24f° 5tJ aA. This aqueduct is constructed of 
brick laid in close mortar joints. Its cross-section is shown in 
Fig. 1266 . It is 13.53 feet in height by 13.6 feet in maximum 
width. The radius of its invert is 18.5 feet, the span of the in¬ 
vert chord is 12.0 feet, and the depression of the invert below the 
chord is 1.0 foot. Its slope is 0.0003. 

Gagings on various portions of the aqueduct of the Jersey 
City Water Supply Company,* a cross-section of which is shown 

in Fig. 1236 , gave, when the 
aqueduct was new, values of the 
coefficient c in the Chezy for¬ 
mula of from 122 to 145, while 
the average value of n in Kut- 
ter’s formula was 0.0127. The 
value of the mean velocity in 
this conduit is closely given by 
the expression v = 13 ir 0 - 50 s 0 50 , 
where s is the observed slope 
of the water surface. This 
slope during the experiments 
varied from 0.00011 to 0.00036, 
the aqueduct being laid on a 
slope of 0.000095. This conduit is of concrete which was 
cast against smooth wooden forms, the invert being made of 
screeded and troweled concrete. 

Owing to the fouling of such conduits as the result of vege¬ 
table growths and the deposition of materials from the water, a 
diminution in capacity of from 10 to 20 percent with age may be 
expected, and accordingly corresponding allowances should be 
made in the design. 

It is to be noted that Kutter’s formula (Art. 118 ) indicates that 
c steadily increases with the hydraulic radius if n and the slope 
be constant. The results of the experiments above quoted, how¬ 
ever, indicate that c becomes constant and has a maximum value 



* By courtesy of Jersey City Water Supply Company, Paterson, N. J. 








Other Formulas for Channels. Art. 124 303 

of not far from 140 for values of the hydraulic radius of 3 feet and 
upward. 

In an aqueduct of masonry constructed so that the water will 
flow in it with a free surface it will be found that the slope of the water 
surface is seldom if ever parallel to the bottom of the aqueduct. 
This, of course, is as it should be, since the expression for the slope is 
x = Q 2 /a 2 c 2 r. Here both a and r vary with Q, and it seldom happens 
that the value of c realized in the completed structure is the same as 
that assumed in the original design. Since the slope of the water 
surface is not parallel to that of the bottom of the aqueduct, there 
results a condition of steady non-uniform flow, and the formula of 
Art. ( 137 ) must be employed whenever precise determinations of the 
value of c are to be made from the results of experiments. 

Prob. 123 . Compute the mean velocity in the New Croton Aque¬ 
duct when it is flowing one-half full. 

Art. 124. Other Formulas for Channels 

Many attempts have been made to express the mean velocity 
and discharge in a channel by the formulas 

v = Cr x s v q = aCr x s v 

where x and y are derived from the data of observations by pro¬ 
cesses similar to those explained in Art. 42 . As a rule these at¬ 
tempts have not proved successful except for special classes of 
conduits, as the exponents of r and s vary with different values 
of r and with different degrees of roughness. For conduits having 
the same kind of surface a formula of this kind may be established 
which will give good results. The values x = § and x = J are 
frequently advocated, y being not far from \; with such values 
C is found to vary less for certain classes of surfaces than the c 
of the Chezy formula, and this seems to be the only strong argu¬ 
ment in favor of exponential formulas. 

Among the many exponential formulas which have been advo¬ 
cated, those derived by Foss may be cited. For surfaces correspond¬ 
ing to Kutter’s values of n less than 0.017 he finds* 

114 6 _8 _6 

r«=Cr 3 s or v = C 11 r lf x xl 
* Journal of Association of Engineering Societies, 1894, vol. 13, p. 295. 


304 


Chap. 9 . Flow in Conduits and Canals 


in which C has the following values: 

for n = 0.009 0.010 0.011 0.012 0.013 0.015 0.017 

C = 23 000 19 000 15 000 12 000 10 000 8 000 6 000 

For surfaces corresponding to Kutter’s values of n greater than 0.018, 
his formula is 

v 2 = Cr* s or v — C~r J s* 

and the values of C for this case are 

for n — 0.020 0.025 0.030 0.035 

C = 5000 3000 2000 1000 

For circular sections running full he also proposes the formula s = 
0.0065 q*i/d 5 . These formulas are open to objection on account of 
the great range in the values of C. 

Tutton*, as the result of a study of many experiments, proposed 
the formula v = Cr ( 1 ' l 7 ~ m) s"\ where s and r represent the slope and 
hydraulic radius as in the Chezy formula. The values of m ranged 
from 0.48 for tarred iron pipes to 0.58 for pipes of lead, tin, and zinc, 
the average for all cases being m = 0.54. Using this value, the for¬ 
mula became „ 0.63 o.m 

v = Cr s 

for which the value of C was given as from 127 to 153 for new cast- 
iron pipes, from 83 to 98 for lap-riveted iron pipes, from 127 to 153 
for wooden pipes, and about 188 for lead, tin, and zinc pipes. 

Williams and Hazen f have discussed experiments on both 
pipes and open channels, and have proposed an exponential for¬ 
mula that is equivalent to 

v = 1.318 cr 0 - 63 * 0 - 54 

in which c has different values for different surfaces and sections, 
but its range of values is less than that of the c of the Chezy 
formula. The values of c and c are the same when r is 1 foot and 
5 is 0.001. The greater the roughness of the surface, the smaller 
is c ; in general, c is supposed to vary but little for different values 
of r. The following shows the range of the mean values of c 
found from the records of experiments with different surfaces: 

* Transactions Engineers’ Society of Western New York, April, 1896. 
f Hydraulic Tables, New York, 1910. 


Other Formulas for Channels. Art. 124 305 


For coated new cast-iron pipes, 
For tuberculated cast-iron pipes, 
For riveted pipes, 

For wooden stave pipes, 

For new wrought-iron pipes, 

For fire hose, rubber lined, 

For masonry aqueducts, 

For brick sewers, 

For plank aqueducts, unplaned, 
For masonry sluiceways, 

For canals in earth, 


from hi to 146 
from 16 to 112 
from 97 to 142 
from 113 to 129 
from 113 to 124 
from 116 to 140 
from 118 to 145 
from 102 to 141 
from 113 to 120 
from 34 to 75 
from 33 to 71 


The authors of this formula suggest that in computations for 
pipe capacity c be taken as 100 for cast-iron, 95 for riveted steel, 
120 for wooden, no for vitrified pipes, 100 for brick sewers, and 
120 for first-class masonry conduits. 

The circumstance that values of C in some of the exponen¬ 
tial formulas of this article have a smaller range of values than 
the c of the Chezy formula is sometimes cited as an argument 
in their favor. While this is a good argument, the fact must 
not be overlooked that probably the true theoretic formula for 
mean velocity in a pipe or channel is of the form noted in the 
first paragraph of Art. 110 . 

In conclusion, it may be noted that when the velocity is very 
low, the Chezy formula is not valid. In such a case the velocity 
does not vary with the square root of the slope, but with its first 
power, the same conditions obtaining as in pipes (Art. 110 ). A 
glacier moving in its bed at the rate of a few feet per year has a 
velocity directly proportional to its slope. Water flowing in a 
channel with a velocity less than one-quarter of a foot per second 
follows the same law, and the formulas of this chapter cannot be 
applied. The formula for this case is v = Cr 2 s, but values of C 
are not known. It is greatly to be desired that series of experi¬ 
ments should be made for determining values of C. 

Prob. 124 ; Compute the fall of the water surface in a length of 1000 
feet for a ditch where v = 3.62 feet per second, r = 2.75 feet, and n = 0.025; 
first by Williams and Hazen’s formula, and second, by formula ( 122 ) and 
Bazin’s coefficients. 


306 


Chap. 9 . Flow in Conduits and Canals 


Art. 125 . Losses of Head 

The only loss of head thus far considered is that due to friction, 
but other sources of loss may often exist. As in the flow in pipes, 
these may be classified as losses at entrance, losses due to curva¬ 
ture, and losses caused by obstructions in the channel or by 
changes in the area of cross-section. 

When water is admitted to a channel from a reservoir or pond 
through a rectangular sluice, there occurs a contraction similar 
to that at the entrance into a pipe, and which may be often ob¬ 
served in a slight depression of the surface, as at D in Fig. 125 a. 


At this point, therefore, the ve¬ 
locity is greater than the mean 
velocity v, and a loss of energy 
or head results from the subse¬ 
quent expansion, which is ap¬ 
proximately measured by the 



Fig. 125 a. 


difference of the depths d\ and d 2 , the former being taken at the 
entrance of the channel, and the latter below the depression 


where the uniform flow is fully established. According to the 


experiments of Dubuat, made late in the eighteenth century, 
the loss of head for this case is 



in which m ranges between o and 2 according to the condition 
of the entrance. If the channel be small compared with the 
reservoir, and both the bottom and side edges of the entrance 
be square, m may be nearly 2; but if these edges be rounded, 
m may be very small, particularly if the bottom contraction is 
suppressed. The remarks in Chap. 5 regarding suppression of 
the contraction apply also here, and it is often important to pre¬ 
vent losses due to contraction by rounding the approaches to 
the entrance. Screens are sometimes placed at the entrance to 
a channel in order to keep out floating matter; if the cross-sec¬ 
tion of the channel is n times that of the meshes of the screen, 
the loss of head, according to ( 76 ) 2 , is (n — 1 )V/ 2 £. 























Losses of Head. Art. 125 


307 


The loss of head due to bends or curves in the channel is small 
if the curvature be slight. Undoubtedly every curve offers a 
resistance to the change in direction of the velocity, and thus 


requires an additional head to cause the flow beyond that needed 
to overcome the frictional resistances. Several formulas have 
been proposed to express this loss, but they all seem unsatisfactory, 
and hence will not be presented here, particularly as the data 
for determining their constants are very 
scant. It will be plain that the loss of 
head due to a curve increases with its 
length, as in pipes (Art. 91 ). When a 
channel turns with a right angle, as in 
Fig. 1256 , the loss of head may be 
taken as equal to the velocity-head, 
since the experiments of Weisbach on such bends in pipes in¬ 
dicate that value. In this case there is a contraction of the 
stream after passing the corner, and the subsequent expansion 
of section and the resulting impact causes the loss of head. 



Fig. 1256 . 


The losses of head caused by sudden enlargement or by sud¬ 
den contraction of the cross-section of a channel may be estimated 
by the rules deduced in Arts. 76 and 77 . In order to avoid these 
losses changes of section should be made gradually, so that energy 
may not be lost in impact. Obstructions or submerged dams 
may be regarded as causing sudden changes of section, and the 
accompanying losses of head are governed by similar laws. The 
numerical estimation of these losses will generally be difficult, 
but the principles which control them will often prove useful 
in arranging the design of a channel so that the maximum work 
of the water can be rendered available. But as all losses of head 
are directly proportional to the velocity-head v 2 /2g, it is plain 
that they can be rendered inappreciable by giving to the channel 
such dimensions as will render the mean velocity very small. 
This may sometimes be important in a short conduit or flume 
which conveys water from a pond or reservoir to a hydraulic 
motor, particularly in cases where the supply is scant, and where 
all the available head is required to be utilized. 





















308 Chap. 9. Flow in Conduits and Canals 

If no losses of head exist except that due to friction, this can 
be computed from ( 113 ) if the velocity v and the coefficient c be 
known. For since the value of s is v 2 /C 2 r and also h/l , where h 
is the fall expended in overcoming friction, h may be found from 

h = ls = lv 2 /c 2 r ( 125 ) 

but this computation will usually be liable to much error. 


As an example of the computations which sometimes occur 
in practice the following actual case will be discussed. From a canal 



A water is carried through a cast-iron pipe B to an open wooden fore¬ 
bay C, where it passes through the orifice D and falls upon an over¬ 
shot wheel. At the mouth of the pipe is a screen, the area between the 
meshes being one-half that of the cross-section of the pipe. The pipe 
is 3 feet in diameter and 32 feet long. The forebay is of unplaned 
timber, 5 feet wide and 38 feet long, and it has three right-angled bends. 
The orifice is 5 inches deep and 40 inches wide, with standard sharp 
edges on top and sides and contraction suppressed on lower side so that 
its coefficient of contraction is about 0.68 and its coefficient of velocity 
about 0.98. The water level in the canal being 3.75 feet above the 
bottom of the orifice, it is required to find the loss of head between 
the points A and D. 

The total head on the center of the orifice is 3.75 — 0.208 = 3.542 
feet. Let vi be the mean velocity in the pipe, v that in the forebay, 
and V that in the contracted section beyond the orifice. The area of 
the cross-section of the pipe is 7.07 square feet; that of the forebay, 
taking the depth of water as 3.7 feet, is 18.5 square feet, and that of 
the contracted section of the jet issuing from the orifice is 0.945 square 
feet. It will be convenient to express all losses of head in terms of the 
velocity-head v 2 /2g, and hence the first operation is to express v l and 
I in terms of v, or V\ = 2.622; and V = 19.62;. Starting with the screen, 
the loss of head due to expansion of section after the water passes 
through it is, by Art. 76 , 

h' = «*)» _ 6 . 9 ^ 


2 g 



































Losses of Head. Art. 125 


309 


The loss of head in friction in the pipe, using 0.02 for the friction factor 
is, by Art. 90 , 


// _ rl Vi 2 v 2 

h r -4 — 

d 2 g 2 g 


The loss of head in the 
bay is, by Art. 76 , 


expansion of section from the pipe 


h' = fa ~ v ) 2 = 

2 g 



to the fore- 


The loss of head in friction in the forebay, taking c from Table 122 
for the hydraulic radius 1.5 feet and degree of roughness m = 0.16, is 
then found to be 



c 2 r 



The loss of head in the three right-angled bends of the forebay is esti¬ 
mated, as above noted, by 


The loss of head on the edges of the orifice is, by Art. 56 , 

T 2 „ v 2 
h =0.041—= 15*9 —■ 

. 2 g ' 2 g 

Now the total head is expended in these lost heads and in the velocity- 
head of the jet issuing from the orifice, or 

v 2 , V 2 v 2 

3.542 = 29.9-|- =417 — 

2g 2g 2g 


from which the value of v 2 /2g is found to be 0.00851 feet. Finally 
the total loss of head or fall in the free surface of the water before 
reaching the orifice is 

v 2 

(29.9 — 15.9) — = 14.0 X 0.00851 = 0.119 feet, 

2 g 

and therefore the water surface at D is 0.119 feet lower than that at 
A , and the pressure-head on the center of the orifice is 3.433 feet. 
This is the result of the computations, but on making measurements 
with an engineer’s level the water surface at D was found to be 0.125 
feet lower than that at A ; the error of the computed result is there¬ 
fore 0.006 feet. 


Prob. 125 . Compute from the above data the velocities v, v\, and V, 
and the discharge through the orifice. Show that the head lost in passing 
through the screen was 0.059 f ee b which is about one-half of the total. 




310 


Chap. 9. Flow in Conduits and Canals 


Art. 126. Velocities in a Cross-section 

For a circular conduit running full and under pressure the 
velocities in different parts of the section vary similarly to those 
in pipes (Art. 86). When it is partly full, so that the water flows 
with a free surface, the air resistance along that surface is much 
smaller than that along the wetted perimeter, and hence the sur¬ 
face velocities are greater than those near the perimeter. Fig. 
126a illustrates the variation of velocities in a cross-section of the 


SCALE OF FEET 

1---- - ■ J - t - 1 ■ ■ 1 

0 12 3 4 



Fig. 126a. 


Sudbury conduit when the water was about 3 feet deep, as deter¬ 
mined by the gagings of Fteley and Stearns.* The 97 dots are 
the points at which the velocities were measured by a current 
meter (Art. 40), and the velocity for each point in feet per second 
is recorded below it. From these the contour curves were drawn 
which show clearly the manner of variation of velocity throughout 
this cross-section. Since the dots are distributed over the area 
quite uniformly, that area may be regarded as divided into 97 
equal parts, in each of which the velocity is that observed, and 
hence the mean of the 97 observations is the mean velocity (Art. 
39). Thus is found z; = 2.620 feet per second, and this is 85 per 
cent of the maximum observed velocity. 

Similarly Fig. 125/; shows the results of an experiment on the 
New Croton Aqueduct.f In this case the average velocity de- 

* Transactions American Society of Civil Engineers, 1883, vol. 12, p. 324. 
f Report of The Aqueduct Commissioners, New York, 1895-1907. 















Velocities in a Cross-section. Art. 126 


311 


termined from the 128 individual observations is 3.570, and this 
is 89 percent of the maximum observed velocity. A description 
of the methods followed in making the gagings on this aqueduct 



is to be found at page 106 of vol. 66, Transactions American So¬ 
ciety of Civil Engineers. See also Art. 123 . 

An examination of the distribution of velocities in Fig. 1266 
indicates that the maximum velocity does not occur at the center 
of the cross-section. This is due to the fact that the aqueduct 
at the point where the gaging was taken is located on a curve 
which tends to throw the maximum velocity away from the 
•center and toward the outside of the curve. 


















































312 


Chap. 9. Flow in Conduits and Canals 


If all the filaments of a stream of water in a channel have the 
same uniform velocity v, the kinetic energy per second of the flow T 
is the weight of the discharge multiplied by the velocity-head ; or 

0)2 J|2 rjZ 

K = W — = wq — = wa — 

2g 2g 2g 


in which W is the weight of the water delivered per second, w 
is the weight of one cubic unit, q the discharge per second, and a 
the area of the cross-section. For this case, therefore, the energy 
of the flow is proportional to the area of the cross-section and to 
the cube of the velocity. Since, however, the filaments have 
different velocities, this expression may be applied to the actual 
flow by regarding v as the mean velocity. To show that this 
method will be essentially correct, Fig. 126a may be discussed, 
and for it the true energy per second of the flow is 

K ' = m(vA + n£ + ... + ®2zT) 

97 W 2 g 2g 1 

now the ratio of this true kinetic energy to the kinetic energy 
expressed in terms of the mean velocity is 

K' _ fli 3 + Z>2 3 + - h^97 3 

K 97 ^ 

By cubing each individual velocity and also the mean velocity, 
there is found K r = 0 . 9992 K, so that in this instance the two 
energies are practically equal, and hence it is probable that in 
most cases computations of energy from mean velocity give 
results essentially correct. 


Prob. 126 . Draw a vertical plane through the middle of Fig. 1266 
and construct a longitudinal vertical section showing the distribution of ve¬ 
locities. Also draw a horizontal plane through the region of maximum ve¬ 
locity and construct a longitudinal horizontal section. Ascertain whether 
the curves of velocity for these sections are best represented by parabolas 
Or by ellipses. 


Art. 127. Computations in Metric Measures 

« 

(Art. 113 ) The coefficient c in the Chezy formula depends upon 
the linear unit of measure. Let c x be the value when v and r are ex¬ 
pressed in feet and c 2 the value when v and r are expressed in meters, 






Computations in Metric Measures. Art. 127 


313 


and let g x and g 2 be the corresponding values of the acceleration of 
gravity. Then since c = V8g//, it is seen that 

c 2 = CiVg2/gi = C1V9. 80/32.16 = 0.552 Ci 

Hence any value of c in the English system may be transformed into 
the corresponding metric value by multiplying by 0.552. The metric 
value of c for conduits and canals usually lies between 16 and 100. 

(Art. 114 ) Table 127 a gives values of the Chezy coefficient c 
for circular conduits, full or half full. In using it a tentative method 
must be employed, and for this purpose there may be used at first, 

mean Chezy coefficient c = 68 

and then, after v has been computed, a new value of c is taken from the 
table and a new v is found. For example, let it be required to find the 
velocity and discharge of a circular conduit of 1.5 meters diameter 
when laid on a grade of 0.8 meters in 1000 meters. First, 

v = 68 X JV1.5 X 0.0008= 1.18 meters per second, 

and for this velocity the table gives about 77 for c. A second compu¬ 
tation then gives v = 1.33 meters per second and from the table c 
is 78.2. With this value is found v = 1.35 meters per second, which 
may be regarded as the final result. When running full, the discharge 
of this conduit is 0.7854 X 1.5 2 X 1.35 = 2.39 cubic meters per second. 

Table 127 a. Chezy Coefficients for Circular Conduits 


Metric Measures 


Diameter 

Velocity in Meters per Second 

in 







Meters 

0.3 

0.6 

0.9 

1 5 

30 

4.5 

0-3 

53 

57 

60 

63 

67 

68 

0-5 

57 

61 

64 

67 

7 i 

73 

0.7 

61 

65 

68 

71 

76 

78 

O.Q 

64 

68 

70 

74 

79 

81 

1.1 

66 

70 

72 

76 

81 

83 

i -3 

68 

72 

74 

78 

83 


1.6 

7 2 

74 

77 

80 



2.0 

74 

77 

79 

83 



2.4 

76 

79 

82 

























314 


Chap. 9 . Flow in Conduits and Canals 


(Art. 115 ) Table 115 is the same for all systems of measures. 
The results in Art. 115 , for Bazin’s semicircular conduits of 1.25 
meters’ diameter on a slope 5 = 0.0015, are as follows, when all dimen¬ 
sions are in meters: 


For cement lining 


For mortar lining 


d' 

r 

V 

c 

d’ 

r 

V 

c 

0.625 

0.314 

1.85 

85 

0.625 

0.312 

1.69 

78 

0.491 

0.264 

1.61 

81 

o.SiS 

0.275 

i- 5 i 

75 

0.314 

0.185 

1.27 

76 

0.332 

0.194 

1.18 

69 

0.180 

0.112 

0.92 

71 

0.186 

0.116 

0.88 

66 


Here the coefficient c for any depth d' may be roughly expressed by 
Ci~$o(\d— d f ), where c x is the coefficient for the conduit half full. 

(Art. 116 ) Table 1276 gives metric values of c for wooden and 
rectangular sections .on a slope 5 = 0.0049, as determined by the work 
of Darcy and Bazin. 


Table 1276 . Chezy Coefficients c for Rectangular 

Conduits 

Metric Measures 


Unplaned Plank 
b = 1.2 Meters 

Unplaned Plank 
b = 2 Meters 

Neat Cement 
b = 1.8 Meters 

Brick 

b = 1.9 Meters 

d 

c 

d 

c 

d 

c 

d 

c 

O.08 

55 

O.06 

49 

O.06 

64 

0.06 

49 

•15 

60 

.09 

56 

.08 

69 

.09 

54 

.18 

61 

A 3 

60 

•13 

73 

A 5 

57 

.27 

63 

.18 

62 

A 7 

74 

A 7 

58 

•30 

63 

.20 

64 

.19 

75 

.20 

58 

.36 

64 

.24 

64 

.21 

75 

.22 

59 

•39 

65 

.27 

65 

.24 

76 

.26 

60 

•44 

65 

.29 

66 

.27 

76- 

•30 

61 


(Art. 117 ) In designing channels in earth the following values 
may be used for preliminary computations: 


for unplaned plank, 

for smooth masonry, 

for clean earth, 

for stony earth, 

for rough stone, 

for earth foul with weeds 


c = 55 to 66 
c = 50 to 61 
c = 33 to 40 
C = 22 to 33 
c = 19 to 28 
c = 17 to 28 






















Computations in Metric Measures. Art. 127 315 

.. V 

(Art. 118 ) When r is in meters and v in meters per second, Kut- 
ter’s formula takes the form 



( 127 )! 


in which the number n depends upon the roughness of the surface, 
its values being those given in Art. 118 . It may be noted that when 
the hydraulic radius r is one meter, the value of c is i/n. 

(Art. 119 ) Metric coefficients for sewers will be found in Table 
127 c. As these are given to the nearest unit only, the error in using 
them is slightly greater than with the larger coefficients of the English 
system. In important cases the values of c may be directly computed 
from Kutter’s formula. 


Table 127 r. Kutter’s Coefficients c for Sewers 


Metric Measures 


Hydraulic 
Radius r 
in Meters 

s = 0.00005 

S = O.OOOI 

s = 

o.ox 

n = 0.015 

n = 0.017 

n = 0.015 

n = 0.017 

n = 0.015 

n = 0.017 

0.05 

26 

22 

31 

25 

37 

30 

O.I 

34 

29 

37 

32 

43 

36 

0.15 

39 

33 

42 

36 

48 

40 

0.2 

43 

38 

46 

40 

5i 

43 

0-3 

49 

42 

5i 

44 

55 

48 

o -5 

56 

48 

57 

50 

60 

52 

0.7 

62 

54 

62 

55 

63 

56 

1.0 

67 

59 

67 

58 

66 

59 


(Art. 120 ) Table 127 d in metric measures corresponds to Table 
120 in English measures and is used in the same manner. 

(Art. 121 ) The metric coefficients c for steel, cast-iron, and wood 
pipes may be obtained from those in the text by multiplying by 0.552, 
while the velocities and diameters may easily be replaced by metric 
equivalents with the help of Table C at the end of this volume. 

(Art. 122 ) The values of c in Table 127 c have been taken from 
the more extended table published in 1897 by Bazin, while those in 

























316 


Chap. 9 . Flow in Conduits and Canals 


Table 122 have been computed by (115). In metric measures Bazin’s 
formula for channels is 

v = cVrs c =-——z (127)2 

1 + ra/W 

in which m has the same values as those given in Art. 122. 

Table 127 d. Kutter’s Coefficients c for Channels 


Metric Measures 


Hydraulic 
Radius r 
in Meters 

5 = 0.00005 

s = O.OOOI 

5 = 

0.0 r 

n = 0.025 

n = 0.030 

n = 0.025 

n = 0.030 

n = 0.025 

n = 0.030 

0.2 

22 

18 

24 

19 

27 

21 

0-3 

27 

22 

29 

33 

31 

25 

0-5 

32 

27 

34 

27 

35 

28 

0.7 

36 

30 

37 

30 

38 

31 

1.0 

40 

33 

40 

33 

40 

33 

i -5 

45 

38 

44 

38 

43 

36 

2. 

48 

41 

47 

40 

45 

38 

3 - 

53 

44 

50 

44 

47 

40 

5 - 

59 

50 

53 

47 

5 i 

43 


Table 127c. Bazin’s Coefficients c for Channels 


Metric Measures 


Hydraulic 
Radius r 
in Meters 

m = 0.06 

m = 0.16 

m = 0.46 

m — 0.85 

m = 1.30 

m = 1.75 

0.2 

76.7 

64.1 

42.9 




0.4 

79-4 

69.4 

504 

37 -i 



0.6 

80.7 

72.I 

54-6 

41.4 

32.5 


0.8 

81.5 

73-8 

574 

44.6 

35-5 

29.4 

1.0 

82.0 

75 -o 

59-6 

47.0 

37-8 

31.6 

1-5 

82.9 

76.9 

63.2 

5 i -3 

42.2 

35-8 

2.0 

83-4 

78.1 

65.6 

54-3 

45-3 

38.9 

2-5 

83.8 

79.0 

67.4 

56.6 

47-7 

41.1 

3 - 

84.0 

79.6 

68.7 

58.3 

49-7 

43-3 

4 - 

84.4 

80.9 

71-5 

61.0 

52.7 

46.4 

5 - 

84.7 

81.2 

72.1 

63.0 

55 -o 

48.8 

6. 


81.6 

73-2 

64.6 

56.8 

50.7 

8. 



74.8 

66.9 

59-5 

53-7 

10. 




68.5 

61.6 

. 56.0 


































Computations in Metric Measures. Art. 127 


317 


(Art. 123) The metric formula for the Sudbury conduit is 
v = 8 o. 9 r°' 6 V 5 , and Foss’ formula Art. 124 for circular conduits 
or large pipes when running full is s = o.oii Sg^ l '/d 5 . 

Prob. 127 u. Compute the value of c for a circular conduit 1.4 meters 
in diameter which delivers 4.86 cubic meters per second when running full, 
its slope being 0.008. 

Prob. 1276 . Find the hydraulic radius for a circular conduit of 1.6 
meters diameter when the water is 1.2 meters deep. 

Prob. 127 c. If the value of c is 30, compute the depth of a trapezoidal 
section to carry 10 cubic meters per second, the slope 5 being 0.0015, the 
bottom width double the depth, and the sides making an angle of 34 0 with 
the horizontal. 

Prob 127 d. A conduit lined with neat cement has a cross-section of 
3.45 square meters and a wetted perimeter of 5.02 meters and its slope is 
0.00025. Compute the discharge in liters per 24 hours, (a) by Kutter’s 
formula, and (6) by Bazin’s formula. 


318 


Chap. 10. The Flow of Rivers 


CHAPTER 10 
THE FLOW OF RIVERS 

Art. 128. General Considerations 

Steady flow in a river channel occurs when the same quan¬ 
tity of water passes each section in each unit of time; here the 
mean velocities in different sections vary inversely as the areas 
of those sections. Uniform flow is that particular case of steady 
flow where the sections considered are equal in area. Uniform 
flow and some other cases of steady flow will be mainly considered 
in this chapter. Non-steady flow occurs when the stage of a 
river is rising or falling, and Art. 134 treats of this case. 

No branch of hydraulics has received more detailed investiga¬ 
tion than that of the flow in river channels, and yet the subject 
is but imperfectly understood. The great object of all these 
investigations has been to devise a simple method of determining 
the mean velocity and discharge without the necessity of expen¬ 
sive field operations. In general it may be said that this end has 
not yet been attained, even for the case of uniform flow. Of 
the various formulas proposed to represent the relation of mean 
velocity to the hydraulic radius and the slope, none has proved 
to be of general practical value except the empirical one of Chezy 
given in the last chapter, and this is often inapplicable on account 
of the difficulty of measuring the slope s and determining the 
coefficient c. The fundamental equations for discussing the 
laws of variation in the mean velocity v and in the discharge q are 

v = c Vrs q = a-c^/rs 

where a is the area of the cross-section and r its hydraulic radius, 
and all the general principles of the last chapter are to be taken 
as directly applicable to uniform flow in natural channels. 


General Considerations. Art. 128 


319 


Kutter’s formula for the value of c is probably the best in the 
present state of science, although it is now generally recognized 
that it gives too large values for small slopes. In using it the 
coefficients for rivers in good condition may be taken from Table 
120 , but for bad regimen n is to be taken at 0 . 03 , and for wild tor¬ 
rents at 0.04 or 0 . 05 . It is, however, too much to expect that a sin¬ 
gle formula should accurately express the mean velocity in small 
brooks and large rivers, and the general opinion now is that 
efforts to establish such an expression will not prove successful. 
In the present state of the science no engineer can afford in any 
case of importance to rely upon a formula to furnish anything 
more than a rough approximation to the discharge in a given 
river channel, but actual field measurements of its velocity must 
be made. 

When these formulas are used to determine the discharge of 
a river, a long straight portion or reach should be selected where 
the cross-sections are as nearly as possible uniform in shape and 
size. The width of the stream is then divided into a number of 
parts and soundings taken at each point of division. The data 
are thus obtained for computing the area a and the wetted perim¬ 
eter p, from which the hydraulic depth r is derived. To determine 
the slope s a length l is to be measured, at each end of which 
bench-marks are established whose difference of elevation is 
found by precise levels. The elevations of the water surfaces 
below these benches are then to be simultaneously taken, whence 
the fall h in the distance l becomes known. As this fall is often 
small, it is very important that every precaution be taken to 
avoid error in the measurements, and that a number of them be 
taken in order to secure a precise mean. Care should be observed 
that the stage of water is not varying while these observations 
are being made, and for this and other purposes a permanent 
gage board must be established. It is also very important that 
the points upon the water surface which are selected for compari¬ 
son should be situated so as to be free from local influences such 
as eddies, since these often cause marked deviations from the 
normal surface of the stream. If hook gages can be used for re¬ 
ferring the water levels to the benches, probably the most accurate 


320 


Chap. 10. The Flow of Rivers 


results can be obtained. It has been observed that the surface 
of a swiftly flowing stream is not a plane, but a cylinder, which is 
concave to the bed, its highest elevation being where the velocity 
is greatest, and hence the two points of reference should be located 
similarly with respect to the axis of the current. In spite of all 
precautions, however, the relative error in h will usually be large 
in the case of slight slopes, unless / be very long, which cannot 
often occur in streams under conditions of uniformity. 

Owing to the uncertainty of determinations of discharge made 
in the manner just described, the common practice is .to gage the 
stream by velocity observations, to which subject, therefore, a 
large part of this chapter will be devoted. The methods given 
are equally applicable to conduits and canals, and in Art. 133 will 
be found a summary which briefly compares the various processes. 

Prob. 128 . Which has the greater discharge, a stream 2 feet deep and 
85 feet wide on a slope of 1 foot per mile, or a stream 3 feet deep and 40 feet 
wide on a slope of 2 feet per mile ? 

Art. 129. Velocities in a Cross-section 

The mean velocity v is the average of all the velocities of all 
the small sections or filaments in a cross-section (Art. 112). Some 
of these individual velocities are much smaller, and others ma¬ 
terially larger, than the mean velocity. Along the bottom of 
the stream, where the frictional resistances are the greatest, the 
velocities are the least; along the center of the stream they are 
the greatest. A brief statement of the general laws of variation 
of these velocities will now be made. 

In Fig. 129 there is shown at A a cross-section of a stream 
with contour curves of equal velocity; here the greatest velocity 
is seen to be near the deepest part of the section a short distance 
below the surface. At B is shown a plan of the stream with ar¬ 
rows roughly representing the surface velocities; the greatest 
of these is seen to be near the deepest part of the channel, while 
the others diminish toward the banks, the curve showing the law 
of variation resembling a parabola. At C is shown by arrows 
the variation of velocities in a vertical line, the smallest being 


Velocities in a Cross-section. Art. 129 


321 


at the bottom and the largest a short distance below the surface; 
concerning this curve there has been* much contention, but it is 
commonly thought to 
be a parabola whose axis 
is horizontal. These are 
the general laws of the 
variation of velocity 
throughout the cross- 
section ; the particular 
relations are of a com¬ 
plex character, and vary so greatly in channels of different kinds 
that it is difficult to formulate them, although many attempts 
to do so have been made. Some of these formulas which con¬ 
nect the mean velocity with particular velocities, such as the 
maximum surface velocity, mid-depth velocity in the *axis of 
the stream, etc., will be given in Art. 132. 



Humphreys and Abbot deduced in 1861 for the Mississippi 
River * an equation of the mean curve of mean velocities in a 
vertical line, nameljq 

V = 3.261 — 0.7922 (y/d ) 2 

in which V is the velocity at any distance y above or below the 
horizontal axis of the parabolic curve and d is the depth of the 
water, the axis being at the distance 0 . 297 ^ below the surface. 
The depth of the axis was found, however, to vary greatly with 
the wind, an up-stream wind of force 4 depressing it to mid-depth, 
and a down-stream wind of force 5.3 elevating it to the surface. 


In a straight channel having a bed of a uniform nature the 
deepest part is near the middle of its width, while the two sides 
are approximately symmetrical. In a river bend, however, the 
deepest part is near the outer bank, while on the inner side the 
water is shallow; the cause of this is undoubtedly due to the 
centrifugal force of the current, which, resisting the change in 
direction, creates currents which scour away the outer bank or 
prevent deposits from forming there. It is well known to ail 


* Physics and Hydraulics of the Mississippi River, edition of 1876, p. 243. 



























322 


Chap. 10. The Flow of Rivers 


that rivers of the least slope have the most bends; perhaps this 
is due to the greater relative influence of such cross currents. 
(See Art. 156.) 

The theory of the flow of water in channels, like that of flow in 
pipes, is based upon the supposition of a mean velocity which is the 
average of all the parallel individual velocities in the cross-section. 
But in fact there are numerous sinuous motions of particles from the 
bottom to the surface which also consume a portion of the lost head. 
The influence of these sinuosities is as yet but little understood ; 
when in the future this becomes known, a better theory of flow in 
channels may be possible. 

Prob. 129 . Show that the above formula for velocities in a vertical 
can be put into the form 

V = 3.19 + 0.471 ( x/d ) — 0.792 (x/d ) 2 

in which x is the depth below the surface. 

Art. 130. Velocity Measurements 

One of the methods for measuring the discharge of streams 
which has been extensively used is by observing the velocity of 
flow by the help of floats. Of these there are three kinds, sur¬ 
face floats, double floats, and rod floats. Surface floats should 
be sufficiently submerged so as to thoroughly partake of the 
motion of the upper filaments, and should be made of such a form 
as not to readily be affected by the wind. The time of their 
passage over a given distance is determined by two observers at 
the ends of a base on shore by stop-watches; or only one watch 
may be used, the instant of passing each section being signaled 
to the time-keeper. If l be the length of the base, and t the time 
of passage in seconds, the velocity of the float is v = l/t. When 
there are many observations, the numerical work of division is 
best done by taking the reciprocals of t from a table and multi¬ 
plying them by /, which for convenience may be an even number, 
such as 100 or 200 feet. 

A sub-surface float consists of a small surface float connected 
by a fine cord or wire with the large real float, which is weighted 
so as to remain submerged and keep the cord reasonably taut. 
The surface float should be made of such a form as to offer but 


Velocity Measurements. Art. 130 


323 


slight resistance to the motion, while the lower float is large, it 
being the object of the combination to determine the velocity of 
the lower one alone. This arrangement has been extensively 
used, but it is probable that in all cases the velocity of the large 
float is somewhat affected by that of the upper one, as well as 
by the friction of the cord. In general the use of these floats is 
not to be encouraged, if any other method of measurement can 
be devised. 

The rod float is a hollow cylinder of tin, which can be weighted 
by dropping in pebbles or shot so as to stand vertically at any 
depth. When used for velocity determinations, they are weighted 
so as to reach nearly to the bottom of the channel, and the time 
of passage over a known distance determined as above explained. 
It is often stated that the velocity of a rod float is the mean 
velocity of all the filaments in contact with it. Theoretically 
this is not the case, but the rod moves a little slower. However, 
in practice a rod cannot reach quite to the bed of the stream, and 
Francis has deduced the following empirical formula for finding 
the mean velocity V m of all the filaments between the surface and 
the bed from the observed velocity V r of the rod : 

V m = V r (i.oi 2 — 0.116 'Vd'/d) 

in which d is the total depth of the stream and d' the depth of 
water below the bottom of the rod.* This expression is probably 
not a valid one, unless d' is less than about one-quarter of d ; 
usually it will be best to have d' as small as the character of the 
bed of the channel will allow. 

The log formerly used by seamen for ascertaining the speed 
of vessels may be often conveniently used as a surface float when 
rough determinations only are required, it being thrown from a 
boat or bridge. The cord of course must be previously stretched 
when wet, so that its length may not be altered by the immersion ; 
if graduated by tags or knots in divisions of six feet, the log may 
be allowed to float for one minute, and then the number of divi- 


* Lowell Hydraulic Experiments, 4th Edition, p. 195. 



324 Chap. 10. The Flow of Rivers 


sions run out in this time will be ten times the velocity in feet 
per second. 

The determination of particular velocities in streams by means 
of floats appears to be simple, but in practice many uncertainties are 
found to arise, owing to wind, eddies, local currents, etc., so that a num¬ 
ber of observations are required to obtain a precise mean result. 

For conduits, canals, and for many rivers the use of a current 
meter will often be found to be more satisfactory and less expensive 
if many observations are required. Comparisons between the re¬ 
sults of float and rod gagings have been made by Murphy.* These 
comparisons include those made at the Cornell University labora¬ 
tory between the weir and the current meter in 1900 . 

Other current indicators less satisfactory for work in streams 
are the Pitot tube and the hydrometric pendulum, shown in Fig. 
130a. The former has not been found valuable for river measure¬ 
ments, although it has proved to be an instrument of great pre¬ 
cision for other classes of work 
(Art. 41), and the latter, although 
used by some of the early hydrau- 
licians, has long been discarded as 
giving only rough indications. The 
same may be said of the hydro¬ 
metric balance, in which weights 
measure the intensity of the pressure of the current, and of the 
torsion balance, in which the pressure of the current on a sub¬ 
merged plate causes the tightening of a spring. These instru¬ 
ments were used only for measurements of velocities in small 
channels, and they are now mere curiosities. 

The current meter, described in Art. 40, is generally operated 
from a bridge or cable in the case of a small stream, but it must 
be often operated from an anchored boat in large rivers. In the 
latter case precise measurements of surface velocities may be 
difficult on account of the eddies around the boat. Even when 
operated from a bridge, it is not easy to obtain successful results 
when the velocity exceeds 4 or 5 feet per second, and special 

* Water Supply and Irrigation Paper No. 95, Washington, 1904. 

























Velocity Measurements. Art. 130 


325 


expedients are necessary to keep the meter in position. How¬ 
ever, the current meter, accurately rated, will in general do bet¬ 
ter work than can be done by floats. 


In using the current meter for the determination of velocity four 
principal methods are used on the work of the U. S. Geological Survey ; 
these have been reviewed by 
Hoyt.* In the first a vertical 
velocity curve is determined by 
placing the meter at regular 
vertical intervals from the sur¬ 
face of the water to the bottom 
of the stream and observing 
the velocity at each such in¬ 
terval. The points so selected 
are usually from io to 20 per¬ 
cent of the water depth apart. 

On plotting the velocities ob¬ 
tained, a curve results which 
graphically indicates the varia¬ 
tions in the velocity as they 
are dependent on the depth. 

The average velocity in the 
vertical can be determined by 
averaging all of the observa¬ 
tions, or more accurately by 
ascertaining the area fixed by the curve and the axis of ordinates 
and then dividing this area by the depth of the water in the ver¬ 
tical. Thus in Fig. 1306 the mean velocity is the area ABC divided 
by the depth 9.5 feet. 



Fig. 1306 . 


In the second of these four methods the velocities at distances 
below the surface of 0.2 and 0.8 of the depth are determined and the 
mean taken as the average velocity in the vertical. Many observa¬ 
tions have proven that this method is correct, and theoretically it 
is based on the mathematical fact that if the velocity curve be a 
parabola, then the mean ordinate will be the average of these at 
points whose abscissas are 0.2114 and 0.7886. 

The third of these methods consists in observing the velocity 


* Transactions American Society of Civil Engineers, 1910, vol. 66. 






















326 


Chap. 10. The Flow of Rivers 


at a distance below the surface equal to 0.6 of the water depth. This 
procedure is also based on the assumption that the velocity curve is 
a parabola whose axis is parallel to the water surface and lies below 
it from o to 0.3 of the water depth. Mathematically, therefore, the 
mean ordinate which represents the mean velocity lies between the 
points whose abscissas are 0.58 and 0.67 of the water depth. 

In the fourth method the mean velocity is determined by observ¬ 
ing the velocity at a point from 0.5 to 1.0 feet below the water 
surface and applying a coefficient determined by observation. This 
coefficient ranges from 0.78 to 0.98, and Hoyt* recommends the fol¬ 
lowing. For average streams in moderate freshets 0.90; during floods 
from 0.90 to 0.95, and for ordinary stages of flow from 0.85 to 0.90. 

In the following tabulation are shown the results obtained in 
476 vertical velocity curves* on 34 rivers in various parts of the United 
States. The depths of these streams ranged from 1.6 to 27.5 feet and 
the observed velocities from 0.25 to 9.59 feet per second. The figures 
given are the coefficients by which the average velocities determined 
by the various methods should be multiplied in order to obtain the 
mean velocity as determined from the vertical velocity curve in the 
first method above described. 


Method 

Coefficient 

Maximum 

Minimum 

Mean 

2 

I.03 

0-95 

O.99 

3 

0.98 

0.79 

0.87 

4 

I.03 

0.97 

I. OO 


In the 476 velocity curves above referred to it was found that the 
point of mean velocity occurred at from 58 to 71 percent of the water 
depth below the surface, and that the average of all the curves showed 
it to be at 0.62 of the depth. 

In cases where the stream to be measured is frozen over it has been 
found that the best work is done by the vertical velocity curve method, 
though the 2 and 8 tenths depth method also gives good results. 
A resume of studies of the flow under ice by Murphy f indicates that 

* Transactions American Society of Civil Engineers, 1910, vol. 66. 
t U. S. Water Supply and Irrigation Paper No. 95, 1904. 











Gaging the Discharge. Art, 131 


327 


the maximum velocity is to be found at from 35 to 40 percent of the 
water depth below the under surface of the ice and that the mean 
velocity occurs at two points, the first from 0.08 to 0.13 and the second 
from 0.68 to 0.74 of the water depth below the under surface of the ice. 

The so-called integration method of determining the average 
velocity in a vertical consists in moving the meter at a uniform rate 
from the surface to the bottom and back again. Each point is thus 
passed over twice, and if all other conditions are the same, the 
average velocity indicated should be the mean velocity in the ver¬ 
tical. This method has, since 1900, come to be practically superseded 
by those before described. 

Prob. 130 . A rod float runs a distance of 100 feet in 42 seconds, the 
depth of the stream being 6 feet, while the foot of the rod is 6 inches above 
the bottom. Compute the mean velocity in the vertical. 

Art. 131 . Gaging the Discharge 

For a very small stream the most precise method of finding 
the discharge is by means of a weir constructed for that purpose. 
Streams of considerable size often have dams built across them, 
and these may also be used like weirs with the help of the coeffi¬ 
cients given in Art. 69 , if there be no leakage through the dam. 
When there are no dams, the method now to be explained is gen¬ 
erally employed. In all cases the first step should be to set up 
a vertical board gage, graduated to feet and tenths, and locate 
its zero with respect to the datum plane used in the vicinity, so 
that the stage of water may at any time be determined by 
reading the gage. 

The place selected for the gaging should be one where the 
channel is free from obstructions and as nearly as possible free 
from bends and curves for some distance both up and down 
stream. One or more sections at right angles to the direction of 
the current are to be established, and soundings taken at inter¬ 
vals across the stream upon them, the water gage being read 
while this is done. The distances between the places of soundings 
are measured either upon a cord stretched across the stream or 
by other methods known to surveyors. The data are thus ob¬ 
tained for determining the areas ai, 0*2, #3, etc., shown upon Fig. 


328 


Chap. 10. The Flow of Rivers 


131 a, and the sum of these is the total area a. Levels should be 
run out upon the bank beyond the water’s edge, so that in case of 

a rise of the stream the ad¬ 
ditional areas can be de¬ 
duced. If a current meter 
is used, but one section is 
needed; if floats are used, 
at least two are required, and these must be located at a place 
where the channel is of as uniform size as possible. 



The mean velocities Vi, v 2 , v 3 , etc., are next to be determined 
for each of the sub-areas. With a current meter this may be done 
by starting at one side of a subdivision, and lowering it at a uni¬ 
form rate until the bottom is nearly reached, then moving it a 
few feet horizontally and raising it to the surface, then moving 
it a few feet horizontally and lowering it, and thus continuing 
until the sub-area has been covered. The velocity then deduced 
from the whole number of revolutions during the time of im¬ 
mersion is the mean velocity for the sub-area. Or, by using any 
one of the methods for determining the mean velocity in the ver¬ 
tical as described in Art. 130 the mean velocity may be deter¬ 
mined. When rod floats are used, they are started above the 
upper section, and the times of passing to the lower one noted, 
as explained in Art. 130 , the velocity deduced from a float at 
the middle of a sub-area being taken as the mean for that area. 
It will be found that the rod floats are more or less affected 
by wind, the direction and intensity of which should always be 
recorded in the field notes. 


The discharge of the stream is the sum of the discharges 
through the several sub-areas, or 

q = aiv i + a 2 v 2 + a 3 v 3 + etc. 

and if this be divided by the total area a , the mean velocity for 
the entire section is determined. 

If di, d 2 , d 3 , etc., are the depths in feet on the several verticals 
in Fig. 131 a, and if v\, v 2 , v 3 , etc., represent the mean velocities 
in feet per second in these verticals, while i is the constant inter- 




























Gaging the Discharge. Art. 131 


329 


val in feet between them, then the discharge in cubic feet per 
second will be given by the formula 

Q = [d\v i + (di + d 2 ) (vi + v 2 ) + d 2 v 2 ] + etc. 
o 


For most cases, however, sufficient 
the expression 


e=* 


'di + do\ fv i + v 2 


accuracy will be given by 
+ etc. 


and this is the method which has been adopted by the U. S. 
Geological Survey. It permits of ready computation, while at 
the same time it does not require absolute uniformity in the 
interval i. Stevens * has compared the various methods and 
formulas which have been used for the computation of the dis¬ 
charge in such cases. 

The following notes give the details of a gaging of the Lehigh 
River, near Bethlehem, Pa., made at low water in 1885 by the use of 
rod floats. The two sections were 100 feet apart, and each was divided 
into 10 divisions of 30 feet width. In the second column are given the 
soundings in feet taken at the upper section, in the third the mean of 
the two areas in square feet, in the fourth the times of passage of the 
floats in seconds, in the fifth the velocities in feet per second, which 
were obtained by dividing 100 feet by the times, and in the last are the 
products a x v h a 2 v 2 , which are the discharges for the subdivisions a h 


a 2) etc. The total discharge is 

found to be 826 cubic feet 

per second, 

Subdivisions 

Depths 

Areas 

Times 

Velocities 

Discharges 

I 

0.0 

55-5 

380 

0.263 

14.6 

2 

3-° 

A 0 

148.5 

220 

0-454 

67.4 

3 

0.0 

201.7 

185 

0.540 

108.9 

4 

7- 1 

217-5 

120 

0.833 

181.2 

5 

7.0 

210.0 

145 

0.690 

144.9 

6 

7.0 

186.0 

150 

0.667 

124.1 

7 

3-3 

150.8 

165 

0.606 

91.4 

8 

4-3 

114.0 

200 

0.500 

57-o 

9 

3-o 

84.0 

320 

0.313 

26.3 

10 


42.0 

430 

O.233 

9-8 


0.0 







a = 1410.0 



q = 825.6 


* Engineering News, June 25, 1908. 








330 


Chap. 10. The Flow of Rivers 


and the mean velocity is v = 826/1410 = 0.59 feet per second. A 
second gaging of the stream, made a week later, when the water 
level was 0.59 feet higher, gave for the discharge 1336 cubic feet per 
second, for the total area 1630 square feet, and for the mean velocity 
0.82 feet per second. 

In the following tabulation are illustrated both the field notes 
and the subsequent computations made to determine the dis¬ 
charge of a stream from a current meter gaging. 


Dis¬ 

tance 

from 

Initial 

Point 

Depth 

of 

Water 

Depth 

of 

Point of 
Obser¬ 
vation 

Time 

in 

Sec¬ 

onds 

Meter 

Rev¬ 

olu¬ 

tions 

Velocity 

Mean 

Water 

Depth 

Dis¬ 

tance 

be¬ 

tween 

Sec¬ 

tions 

Area of 
Section 

Dis¬ 
charge 
Cubic 
Feet per 
Second 

At 

Point 

Mean 

in 

Ver¬ 

tical 

Mean 

in 

Sec¬ 

tion 

O 

O 


— 

— 

O 

O 







- 

0.4 

33 

40 

2.80 


I.08 

1.0 

3 -o 

3 -o 

3-2 

3 

2 

1.6 

61 

40 

I.52 

2.16 








1.2 

36 

50 

3.20 


2.40 

4.0 

2.0 

8.0 

19.2 

5 

6 

4.8 

45 

40 

2.08 

2.64 








2.0 

37 

60 

3 - 7 o 


2-95 

8.0 

5 -o 

40.0 

118.0 

10 

10 

8.0 

41 

50 

2.82 

3.26 








1.0 

49 

70 

3-30 


3-05 

7-5 

5 -o 

37-5 

114.4 

15 

5 

4.0 

39 

40 

2.38 

2.84 








0.6 

32 

40 

2.90 


2.60 

4.0 

3 -o 

12.0 

31.2 

18 

3 

2.4 

44 

80 

1.82 

2.36 













1.18 

i -5 

2.0 

3 -o 

3-5 

20 

0 

— 



0 

O 













Totals 

20.0 

103-5 

289.5 


After a number of discharge measurements have been made 
at a particular gaging station and at a number of different gage 
heights or water stages it becomes possible to plot for the station 
a rating curve which will show the discharge of the stream at 
any given stage. It is also convenient, for purposes of record and 
comparison, to plot on the same sheet the curves of mean velocity 
and areas of the cross-sections for each of the gagings. As new 
measurements of discharge are made they, together with their 
corresponding velocities and areas, may then be plotted upon this 
sheet, and any errors or differences such as those due to a change 
in the stream bed become at once apparent. Such curves are 
shown in Fig. 1316 . 



































Gaging the Discharge. Art. 131 


331 


After the discharge curve for a station has been established, it 
becomes possible, by keeping a record of the gage heights at the 
station, to determine the total quantity of water which passes 
the station in any given time. Observations on the gage height 



Discharge in cubic feet per second 
Fig. 131ft. 


may be made from two to three or more times a day, and in cases 
where the highest accuracy is desired a self-recording gage, such 
as described in Art. 34 , should be installed for the purpose of 
getting a continuous record of the water height. 

When changes in the bed of a stream occur as the result of 
scouring during freshets, or from the formation of bars or other 
causes, new rating curves must be constructed, and care should 
always be taken to see that all of the water flowing down the 
stream passes the section at which measurements are made. 
If diversions past the point of gaging occur, or in case two or 
more channels are found during times of high water, proper 
allowances or new gagings should be made. 

As to the accuracy of the above described methods of gaging 
the discharge it may be said that with ordinary work, using rod 
floats, the discrepancies in results obtained under different con¬ 
ditions ought not to exceed io percent; and with careful work, 
using current meters, they may often be of a higher degree of 





























































332 


Chap. 10. The Flow of Rivers 


precision. In any event the results derived from such gagings 
are more reliable than can be obtained by the use of any formula 
for the discharge of a stream. 

Prob. 131 . A stream 140 feet wide is divided into seven equal parts, 
the six soundings being 1.9, 4.0, 4.8, 4.6, 2.7, and 1.0 feet. The seven veloci¬ 
ties as found by a current meter are 0.7, 1.6, 2.4, 3.5, 3.0, 1.4, and 0.6 feet 
per second. Compute the discharge. 

Art. 132 . Approximate Gagings 

When the mean velocity v of a stream can be found, the dis¬ 
charge is known from the relation q = av, the area a being meas¬ 
ured as explained in the last article. An approximate value of 
v may be ascertained by one or more float measurements by means 
of relations between it and the observed velocity of the floats 
which have been deduced by the discussion of observations. Such 
measurements are usually less expensive than those explained 
in Art. 131 , and often give information which is sufficient for the 
inquiry in hand. 

The ratio of the mean velocity v to the maximum surface 
velocity V has been found to usually lie between 0.7 and 0.85, 
and about 0.8 appears to be a rough mean value. Accordingly, 

v = 0.8 V 

from which, if V be accurately determined, v can be computed 
with an uncertainty usually less than 20 percent. Many at¬ 
tempts have been made to deduce a more reliable relation be¬ 
tween v and V. The following rule derived from the investiga¬ 
tions of Bazin makes the relation dependent on the coefficient c, 
the value of which for the particular stream under consideration 
is to be obtained from the evidence presented in the last chapter: 



It is probable, however, that the relation depends more on the 
hydraulic radius and the shape of the section than upon the degree 
of roughness of the channel, which c mainly represents. 

The influence of wind upon the surface velocities is so great 
that these methods of determining v may not give good results 


Approximate Gagings. Art. 132 


333 


except in calm weather. A wind blowing up-stream decreases 
the surface velocities, and one blowing down-stream increases 
them, without materially affecting the mean velocity and dis¬ 
charge of the stream. 

The ratio of the mean velocity v\ in any vertical to its surface 
velocity V x is less variable, for it lies between 0.79 and 0.98, or 

Vi = 0.86 Vi 

may be used with but an uncertainty of a few per cent. If sev¬ 
eral velocities V x , V 2 , etc., are determined by surface floats, the 
mean velocities v 2 , etc., for the several sub-areas a x , a 2 , etc., 
are known, and the discharge is q = a x v 1 -f- a 2 v 2 -f etc., as before 
explained. 

By means of a sub-surface float, or by a current meter, the 
velocity V' at mid-depth in any vertical may be measured. The 
mean velocity V\ in that vertical is very closely 

V\ = 0.98 V' 

In this manner the mean velocities in several verticals across 
the stream may be determined by a single observation at each 
point, and these may be used, as in Art. 131, in connection with 
the corresponding areas to compute the discharge. 

It was shown by the observations of Humphreys and Abbot 
on the Mississippi that the velocity V' is practically unaffected 
by wind, the vertical velocity curves for different intensities of 
wind intersecting each other at mid-depth. The mid-depth 
velocity is therefore a reliable quantity to determine and use 
in approximate gagings, particularly as the corresponding mean 
velocity v x for the vertical rarely varies more than 1 or 2 per cent 
from the value 0.98 V'. 

Since the maximum surface velocity is greater than the mean 
velocity v, and since the velocities at the shores are usually small, it 
follows that there are in the surface tw r o points at which the velocity 
is equal to v. If by any means the location of either of these could be 
discovered, a single velocity observation would directly give the value 
of v. The position of these points is subject to so much variation 
in channels of different forms, that no satisfactory method of locat¬ 
ing them has yet been devised. 


334 


Chap. 10. The Flow of Rivers 


In cases where it is desired to construct an approximate discharge 
curve and where only a few discharge measurements have been made, 
the method indicated by Stevens* may be followed. From a cross-sec¬ 
tion of the stream the values of a vV in the Chezy formula q— ac vVs 
may be determined for each gage height and a curve plotted. The 
discharge q then being known for several gage heights, it becomes pos¬ 
sible to determine a value for cV^. The value of this latter function 
is nearly a constant, and the desired discharge curve can thus be ap¬ 
proximated. 

Other methods of making approximate gagings consist in adding 
a solution of some chemical or salt to the water of the stream to be 
measured at some point where thorough mixing will occur. If the 
strength of the chemical solution and the rate of its application are 
known, and if samples of the water of the stream are taken above 
the point where the solution is introduced and down-stream after 
thorough mixing has occurred, the discharge of the stream is then 
equal to the number of times the chemical solution has been diluted by 
the water of the stream multiplied by the rate of application of the 
chemical. For example, if 2 quarts of a solution of common salt con¬ 
taining 10 000 parts per million of chlorine be added each second to 
the stream and if a sample taken one-half a mile down-stream shows 
the chlorine to be 20 parts per million then the dilution has been 
10 000/20 or 500 and the discharge then is 500 X 2 quarts = ioco 
quarts per second. No account has here been taken of the chlorine 
naturally found in the water of the stream, and this must in all 
cases be allowed for. Stromeyerf has experimented in this manner 
with solutions of common salt and sulphuric acid. On small streams 
he found that the results agreed well with both the measurements of 
a weir and a Venturi meter, thus leading him to conclude that results 
correct within 1 percent can be obtained in this manner. It is doubt¬ 
ful, however, if such accuracy could be had in large streams. 

Benzenberg,J in gaging the flow in a portion of the sewer system 
of Milwaukee where the sewer lay in a tunnel below the hydraulic 
gradient, injected a quantity of red eosine into the water at one end 
of the tunnel and observed its appearance at the other. He found 
that the color in the water was never distributed over a length 

* Engineering News, July 18, 1Q07. 

t Proceedings, Institution of Civil Engineers, vol. 160. 

t Transactions American Society of Civil Engineers, December, 1893. 


Comparison of Gaging Methods. Art. 133 


335 


greater than 7 to 9 feet, and thus the mean velocity was determined 
with great accuracy. This experiment was of interest also in indi¬ 
cating the relatively small extent to which the particles of water in a 
given cross-section, such as that of a sewer, become separated from 
each other, even during a one-half mile journey. 

Prob. 132 . A stream 60 feet wide is divided into three sections, having 
the areas 32, 65, and 38 square feet, and the surface velocities near the middle 
of these are found to be 1.3, 2.6, and 1.4 feet per second. What is the ap¬ 
proximate mean velocity of the stream and its discharge ? 

Art. 133 . Comparison of Gaging Methods 

This chapter, together with those preceding, furnishes many 
methods by which the quantity of water flowing through an orifice, 
pipe, or channel may be determined. A few remarks will now be 
made by way of summary and comparison. 

The method of direct measurement in a tank is always the 
most accurate, but except for small quantities is expensive, and 
for large quantities is impracticable. Next in reliability and con¬ 
venience come the methods of gaging by orifices and weirs. An 
orifice one foot square under a head of 25 feet will discharge about 
24 cubic feet per second, which is as large a quantity as can 
usually be profitably passed through a single opening. A weir 
20 feet long with a depth of 2.0 feet will discharge about 200 cubic 
feet per second, which may be taken as the maximum quantity 
that can be conveniently thus gaged. The number of weirs may 
be indeed multiplied for larger discharges, but this is usually 
forbidden by the expense of construction. Hence, for larger 
quantities of water indirect measurements must be adopted. 

The formulas deduced for the flow in pipes and channels in 
Chaps. 8 and 9 enable an approximate estimation of their dis¬ 
charge to be determined when the coefficients and data which 
they contain can be closely determined. The remarks in Art. 
128 indicate the difficulty of ascertaining these data for streams, 
and show that the value of the formulas lies in their use in cases 
of investigation and design rather than for precise gagings. For 
pipes an accurately rated water meter is a convenient method of 


336 


Chap. 10 . The Flow of Rivers 


measuring the discharge, while for conduits it will often be found 
difficult to devise an accurate and economical plan for precise 
determinations, unless the conditions are such that the discharge 
may be made to pass over a weir or to be retained in a large 
reservoir, the capacity of which is known for every tenth of a foot 
in depth. For large aqueducts, and for canals and streams, 
the usually available methods are those explained in this chapter. 
In the case of the Catskill Aqueduct under construction in 1912 
a number of Venturi meters of capacities up to 770 cubic feet 
per second have been introduced (Art. 39 ). 

Surface floats are not to be recommended except for rude 
determinations, because they are affected by wind and because 
the deduction of mean velocities from them is in many cases 
subject to much uncertainty. Nevertheless many cases arise in 
practice where the results found by the use of surface floats are 
sufficiently precise *to give valuable information concerning the 
flow of streams. The double float for sub-surface velocity 
is used in deep and rapid rivers, where a current meter can¬ 
not be well operated on account of the difficulty of anchoring 
a boat. In addition to its disadvantages already mentioned may 
be noted that of expense, which becomes large when many ob¬ 
servations are to be taken. 

The method of determining the mean velocities in vertical 
planes by rod floats is very convenient in canals and channels 
which are not too deep or too shallow. The precision of a veloc¬ 
ity determination by a rod float is always much greater than that 
of one taken by the double float, so that the former is to be pre¬ 
ferred when circumstances will allow. In cases where the velocity 
is rapid, or where there are no bridges over the stream, rod floats 
may often give results more reliable than can be obtained by 
any other method. 

Current-meter observations are those which now generally 
take the highest rank for precision in streams where the condi¬ 
tions are not abnormal. The first cost of the outfit is greater than 
that required for rod floats, but if much work is to be done, it will 
prove the cheaper. The main objection is the difficulty of use 


Variations in Discharge. Art. 134 


337 


in cases of high velocities and to the errors which may be intro¬ 
duced from the lack of proper rating; this is required to be done 
at intervals, since it is found that the relation between the velocity 
and the recorded number of revolutions may change during use. 

In the execution of hydraulic operations which involve the meas¬ 
urement of water a method is to be selected which will give the highest 
degree of precision with given expenditure, or which will secure a 
given degree of precision at a minimum expense. Any one can build 
a road, or a water-supply system; but the art of engineering teaches 
how to build it well, and at the least cost of construction and main¬ 
tenance. Similarly the science of hydraulics teaches the laws of How 
and records the results of experiments, so that when the discharge of 
a conduit is to be measured or a stream is to be gaged, the engineer 
may select that method which will furnish the required information 
in the most satisfactory manner and at the least expense. 

Prob. 133 . Consult Humphreys and Abbot’s Physics and Hydraulics 
of the Mississippi River (Washington, 1862 and 1876), and find two methods 
of measuring the velocity of a current different from those described in the 
preceding pages. 

Art. 134 . Variations in Discharge 

When the stage of water rises and falls, a corresponding in¬ 
crease or decrease occurs in the velocity and discharge. The 
relation of these variations to the change in depth may be approx¬ 
imately ascertained in the following manner, the slope of the 
water surface being regarded as remaining uniform: Let the 
stream be wide, so that its hydraulic radius is nearly equal to 
the mean depth d ; then 

v = c^/ds = c 

Differentiating this with respect to v and d gives 

Sv/v = J hd/d 

Here the first member is the relative change in velocity when the 
depth varies from d to d ± Bd, and the equation hence shows that 
the relative change in velocity is one-half the relative change in 
depth. For example, a stream 3 feet deep, and with a mean 
velocity of 4 feet per second, rises so that the depth is 3.3 feet; 


338 


Chap. 10. The Flow of Rivers 


thendz> = 4X^Xo.3/3=o.2, and the velocity of the stream be¬ 
comes 4+ o.2 = 4.2 feet per second. 

In the same manner the variation in discharge maybe found. 
Let b be the breadth of the stream, then 

q = c bd \^ds = c bs 2 d~ 

and by differentiating with respect to q and d , 

hq/q = 2 &d/d 

Hence the relative change in discharge is if times that of the 
relative change in depth. This rule, like the preceding, supposes 
that Sb is very small, and will not apply to large variations in 
the depth of the water. 

The above conclusions may be expressed as follows: If the 
mean depth changes 1 percent, the velocity changes 0.5 percent, 
and the discharge changes 1.5 percent. They are only true 
for streams with such cross-sections that the hydraulic radius 
may be regarded as proportional to the depth, and even for 
such sections are only exact for small variations in d and v. 
They also assume that the slope 5 remains the same after 
the rise or fall as before; this will be the case if a condition of 
permanency is established, but, as a rule, while the stage of 
water is rising the slope is increasing, and while falling the slope 
is decreasing. 

Gages for reading the stages of water are now set up on many 
rivers, and daily observations are taken. Such a gage is usually 
a vertical board graduated to feet and tenths and set if possible 
with its zero below 7 the lowest knowm water level. Another form 
is the box-and-chain gage, which consists of a box fastened on a 
bridge with a graduated scale within it and a chain that can be 
let down to the w 7 ater level; the length of the chain being known, 
the gage height can then be read from the scale if its zero is set 
so that the reading will be zero w r hen the end of the chain just 
touches the water surface when it is at zero height. Such ob¬ 
servations of the daily stage of a river are of great value in plan¬ 
ning engineering constructions, and they are now made at many 


Transporting Capacity of Currents. Art. 135 339 

stations by the United States government through the Depart¬ 
ment of Agriculture and the Geological Survey Bureau. 

When several measurements of the discharge of a stream have 
been made for different stages of water, a curve may be drawn 
to show the law of variation of discharge (Art. 131 ), and from 
this curve the discharge corresponding to any given stage of water 
may be approximately ascertained. Fig. 1316 shows a typical 
discharge curve. Fig. 134 shows the actual discharge curve for 
the Lehigh River at Bethlehem, Pa., the ordinates being the 



0 5000 10000 15000 30000 


Fig. 134 . 

heights of the water level as read on the gage, and the abscissas 
being the discharges of the river in cubic feet per second; this is 
only a part of the discharge curve for that river, as the water has 
been known to rise to 22.5 feet and the corresponding discharge 
was over 100 000 cubic feet per second. Each station on a river 
has its own distinctive discharge curve, for the local topography 
determines the heights to which the water level will rise. 

Prob. 134. A stream of 4 feet mean depth delivers 800 cubic feet per sec¬ 
ond. What will be the discharge when the depth is decreased to 3.87 feet ? 
If the stream is 100 feet wide, what will be the velocity when the depth is 
4.12 feet ? 

Art. 135 . Transporting Capacity of Currents 

The fact that the water of rapid streams transports large 
quantities of earthy matter, either in suspension or by rolling it 
along the bed of the channel, is well known, and has already been 
mentioned in Art. 120 . It is now to be shown that the diameters 
of bodies which can be moved by the pressure of a current vary 
as the square of its velocity, and that their weights vary as the 
sixth power of the velocity. 









































340 


Chap. 10 . The Flow of Rivers 


When water causes sand or pebbles to roll along the bed of a 
channel, it must exert a force approximately proportional to the 
square of the velocity and to the area exposed (Art. 27 ), or if d 
is the diameter of the body and C a constant, the force which is 
required to move it horizontally is 

F = Cd 2 v 2 

But if motion just occurs, this force is also proportional to the 
weight of the body, because the frictional resistance of one body 
upon another varies as the normal pressure or weight. And as 
the weight of a sphere varies as the cube of the diameter, it follows 

d 3 = Cd 2 v 2 or d — Cv 2 

Now since d varies as v 2 , the weight of the body, which is propor¬ 
tional to d 3 , must vary as v 6 ; which proves the proposition enun¬ 
ciated above. Hence an increase in velocity causes far greater 
increase in transporting capacity. 

Since the weight of sand and stones when immersed in water 
is only about one-half their weight in air, the frictional resistances 
to their motion are slight, and this helps to explain the circum¬ 
stance that they are so easily transported by currents of moderate 
velocity. It is found by observation that a pebble about one 
inch in diameter is rolled along the bed of a channel when the 
velocity is about 3J feet per second ; hence, according to the above 
theoretical deduction, a velocity five times as great, or 17J feet 
per second, will carry along stones of 25 inches diameter. This 
law of the transporting capacity of flowing water is only an ap¬ 
proximate one, for the recorded experiments seem to indicate 
that the diameters of moving pebbles on the bed of a channel do 
not vary quite as rapidly as the square of the velocity. The law, 
moreover, is applicable only to bodies of similar shape, and 
cannot be used for comparing round pebbles with flat spalls. 
The following table gives the velocities on the bed or bottom of 
the channel which are required to move the materials stated. 
The corresponding mean velocities in the last column are derived 
from the empirical formula deduced by Darcy, 

v = v -\-11 V rs 


Transporting Capacity of Currents. Art. 135 


341 


in which v' is the bottom and v the mean velocity. The bottom 
or transporting velocities were deduced by Dubuat from experi¬ 
ments in small troughs, and hence are probably slightly less than 
the velocities which would move the same materials in channels 
of natural earth. 



Bottom 

velocity 

Mean 

velocity 

Clay, fit for pottery, 

0-3 

0.4 

Sand, size of anise-seed, 

0.4 

o-S 

Gravel, size of peas, 

0.6 

0.8 

Gravel, size of beans, 

1.2 

1.6 

Shingle, about 1 inch in 

diameter, 2.5 

3-5 

Angular stones, about 1 

| inches, 3.5 

4-5 

The general conclusion to be derived from these figures is that 
ordinary small, loose earthy materials will be transported or rolled 

along the bed of a channel by velocities of 2 or 

3 feet per second. 


It is not necessarily to be inferred that this movement of the 
materials is of an injurious nature in streams with a fixed regimen, 
but in artificial canals the subject is one that demands close at¬ 
tention. The velocity of the moving objects after starting has 
been found to be usually less than half that of the current.* 

In a silt-bearing stream there is a certain critical velocity Vo at 
which all silt already in suspension is carried on without being de¬ 
posited and at which no further silt is scoured from the sides and 
bottom. This velocity, according to the investigation of Kennedy,f 
is given by Vo = md 0M where d is the depth of the stream and m is 
0.82 for light sandy silt, 0.99 for sandy loam, and 1.07 for coarse silt. 
Kennedy also found that the amounts of silt carried in the same 
stream varied with the square root of the fifth power of the veloci¬ 
ties, so that if x and x 0 are amounts carried at velocities V and Vo 

5 

then x = xo (F/Fo)*. When V is greater than Fo, then 3; — x 0 is the 
amount of scour due to the change of velocity; when V is less 
than Vo, then x 0 — x is the amount of deposit due to the change of 
velocity. 

Prob. 135 . In the early history of the earth the moon was half its 
present distance from the earth’s center, and the tides were about eight times 

* Herschel, on the erosive and abrading power of water, in Journal 

Franklin Institute, 1878, vol. 75, p. 330. 

f Proceedings, Institution of Civil Engineers, vol. 119, 1894-95, p. 281. 


342 


Chap. 10. The Flow of Rivers 


as high as at present. It is supposed that these tides rolled over the low 
lands and moved great rocks from place to place. The greatest velocity 
of such a wave is V gd , where d is the depth of the water. What is the prob¬ 
able weight and size of the largest rock that such a current would move ? 

Art. 136 . Influence of Dams and Piers 

When a dam is built across a stream, it is often desired to 
compute its height so that the water level may stand at a given 
elevation. Thus in the figures, CC represents the surface of the 
stream before the construction of the dam, the depth of the water 
being D, and it is required to find the height G of the dam so that 




the water surface may be raised the vertical distance d. There 
are two cases, the first where the crest is above the original water 
level CC, and the second where it is below that level; in both 
cases the discharge q must be known in order to compute the height 
of the dam. 

When the crest is not submerged, as in Fig. 136 #, it is seen that 
the value of G is D + d — H, where H is the head on the crest. 
Now from Art. 64 the value of q is m b(H + i \h)^, where b is the 
length of the crest and h is the head due to velocity of approach. 
Hence there results 

G = D-\-d-\- i\h — (q/uby ( 136 )i 

in which M is to be taken from Art. 69 . For example, let the 
discharge be 18 ooo cubic feet per second, let the width of the 
stream above the dam be 6oo feet, and the width on the crest 
be 525 feet; also let D and d be 8.5 and 6.0 feet, and let m be 
3.33. The mean velocity of approach is 

18 000 f , 

v = - -= 2.1 teet per second 

600 X 14.5 

whence the \elocit\ head is h — 0.0155 X 2.i 2 = 0.07 feet. 
Then from the formula there results G = 9.9 feet, which is the 






























Influence of Dams and Piers. Art. 136 


343 


required height of the dam. In many cases it will be unnecessary 
to consider velocity of approach, and h may be omitted from the 
formula; if this be done for the example in hand, the value of G 
is 9.8 feet. 

When it is desired to raise the water level only a short distance, 
the crest of the dam will be submerged. For this case Fig. 1366 
gives H = D -f d — G and H' = D — G. By inserting these 
heads in formula ( 67) 2 and neglecting velocity of approach, there 
is found 

G = D+id~iq/ubVd ( 136 )* 


Here the coefficient m lies between 3.09 and 3.37, depending on the 
value of the ratio H'/H, and as a mean 3.1 may be used. For ex¬ 
ample, let q = 400 cubic feet per second, D = 4, d=i,b = 50 
feet; then G is found to be 2.95 feet. The value of H is then 
2.05 feet and that of H' is 1.05, whence H'/H is 0.5 closely, and 
from Art. 67 the value of m is 3.11, which indicates that the as¬ 
sumed value is close enough. Accordingly 3.0 feet may be taken 
as the height of the submerged dam. 


Hr 
1 . 




1 

1 — 





When bridge piers are built in a stream, its cross-section is 
diminished and the water level up-stream from the piers stands 
at a greater height than be¬ 
fore. The most common 
problem is to find how high 
the water will rise when the 
original width B is to be con¬ 
tracted to the width b. Let 
D (Fig. 136 c) be the mean 
depth of the water before the 
building of the piers, H the 
rise in the water level, and q the discharge of the stream. Then 
the discharge q may be regarded as consisting of two parts, first 
that passing over a weir of breadth B under the head H, and 
second that passing through the submerged orifice of breadth b 
and height D under the head H. Hence, from Arts. 64 and 51 , 



/////i///////////// /////J//^ //A// /* 

Fig. 136 c. 


c v^(|5 (H + h)i + bD (H + h)i) = q 


(136), 


















344 


Chap. 10. The Flow of Rivers 


in which h is the head due to the velocity of approach. The 
coefficient of discharge c for weirs and orifices is about 0.6, but 
here it is much larger, since there is no crest. From experiments 
by Weisbach on a small round pier, c appears to be over 0.9, and 
from other discussions it appears in some cases to be a little lower 
than 0.8. Its value in any event depends upon the shape of the 
piers and their cutwaters, and probably the best that can now 
be done is to take it as 0.9 for piers with round ends and at 0.8 
for piers with triangular cutwaters. 

As an example of the determination of c, take the case of a 
flood in the Gungal River,* where B = 650, b = 578, and D = 35 
feet and q = 477 800 cubic feet per second, and where it was ob¬ 
served that the height Ii was 3.6 feet. The mean velocity above 
the piers was v = 477 800/38.6 X 650 = 19.0 feet per second, 
whence the velocity-head h = 5.61 feet. Inserting all these data 
in the formula and solving for c, there is found c = 0.79. This 
is an unusual case where the velocity was very high, and the piers 
had sharp cutwaters. 

As an example of the determination of the height H , take the 
case of a bridge over the Weser,f where B = 593, b = 315, D — 
16.4 feet, and q = 46 550 cubic feet per second. As nothing is 
known about the shape of the piers, c may be taken as 0.8; then 
formula (136)3 reduces to 

(II + h) 2 + 13.1 (H -\-hy = 18.3 

from which H + h is found by trial to be 1.55 feet. Now, as¬ 
suming II as 1.2 feet, the mean velocity above the piers is found 
to be 4.3 feet per second, whence li is 0.29 feet. Accordingly 
II = 1.55 — 0.29 = 1.26 feet, and with this value the velocity 
above the pier is found to be 4.44 feet per second, whence a better 
value of h is 0.31 feet. This gives II = 1.24 feet, which may be 
regarded as the final result for the height of the backwater. 

Prob. 136 . A river 940 feet wide has a mean depth of 4.1 feet and a mean 
velocity of 3.3 feet per second. Ten piers, each 12 feet wide, are to be built 

* Proceedings, Institution of Civil Engineers, 1868, vol. 27, p. 222. 
f D’Aubuisson’s Treatise on Hydraulics, Bennett’s translation (New 

York, 1857), p. 189. 


Steady Non-uniform Flow. Art. 137 


345 


across it. Compute the probable rise of backwater caused by the piers. 
Compute also the probable rise during a flood which increases the mean depth 
to 18.5 feet and the mean velocity to 5.8 feet per second. 


Art. 137 . Steady Non-uniform Flow 


In Arts. 112-133 the slope of the channel, its cross-section, 
and its hydraulic radius have been regarded as constant. If 
these are variable in different reaches of the stream, the case is 
one of non-uniformity, and this will now be discussed. The flow 
is still regarded as steady, so that the same quantity of water 
passes each section per second, but its velocity and depth vary 
as the slope and cross-section change. Let there be several 
reaches 4, l 2 , ••• /», which have the falls hi, h 2 , ••• h n , the water 
sections being a\, a 2 , ••• a n , the hydraulic radii r\, r 2 , ••• r n , and 
the velocities V\, v 2 , ••• v n . The total fall hi + hz + ••• + h n 
is expressed by h. Now the head corresponding to the mean 
velocity in the first section is Vi/ 2 g. The theoretic effective head 
for the last section is h + Vi/2g, while the actual velocity-head 
is Vn/2g. The difference of these is the head lost in friction; 
or by ( 125 ), 


h + 




l\v 1 , I2V2 , 

2 I 9 — r 

C1V1 c 2 r 2 


+ 


InVn 

Un Vn 


in which C1 2 , c 2 2 , ••• c n 2 are the Chezy coefficients for the dif¬ 
ferent lengths. Now let q be the discharge per second; then, 
since the flow is steady, the mean velocities are 

Vi = q/ai v 2 = q/a 2 ••• v n = q/a n 


and, inserting these in the equation, it reduces to 


h 


q 2 / 1 


2g \a n 2 ai 


\ + ? 


h 


v c{ 2 ai 2 ri 


- + 


h 


c 2 a 2 r 2 


+ ... + 


In 


C 2 Y 

U'n ' n / 


which is a fundamental formula for the discussion of steady flow 
through non-uniform channels. This formula shows that the dis¬ 
charge q is a consequence not only of the total fall h in the 
entire length of the channel, but also of the dimensions of the 
various cross-sections. The assumption has been made that a 
and r are constant in each of the parts considered; this can be 











346 


Chap. 10. The Flow of Rivers 


realized by taking the lengths h, l 2 , ••• l n sufficiently short. If 
only one part be considered in which a and r are constant, a n and 
ai are equal, all the terms but one in the second member disappear, 
and the last equation reduces to q = ca ^rh/l,which, is the Chezy 
formula for the discharge in a channel of uniform cross-section. 

An important practical problem is that where the steady flow 
is non-uniform in a channel having a bed with constant slope, a 
condition which may be caused by an obstruction below the part 
considered or by a sudden fall below it. Let a\ and a 2 be the 
areas of the two sections, l their distances apart, and Vi and v 2 
the mean velocities. Then, if a and r be average values of the 
areas and hydraulic radii of the cross-sections throughout the 
length /, the last formula becomes 

*= 2 -(A--,+ 44 -) 

2 g \a 2 up c z a~rJ 


Now the important problem is to discuss the change in depth 
between the two sections. For this purpose let A\A 2 in Fig. 137 

be the longitudinal profile of the 
water surface, let A\D be hori¬ 
zontal, and AiC be drawn parallel 
to the bed B\B 2 . The depths 
AiBi and A 2 B 2 are represented 
by di and d 2 , the latter being 
taken as the larger. Let i be the 
constant slope of the bed B\B 2 , then DC = il , and since DA 2 = h 
and A 2 C = d 2 — d \, there is found for the fall in the length /, 



h = il — {d 2 — di) 


Inserting this value of h in the preceding equation and solving 
for /, there is obtained the important formula 


1 = 


2g \ai a 2 J 


( 137 ): 


i - q 2 /c 2 a 2 r 


from which the length l corresponding to a change in depth d 2 — d i 
can be approximately computed. This formula is the more 
accurate the shorter the length l, since then the mean quantities 








Steady Non-uniform Flow. Art. 137 


347 


a and r can be obtained with greater precision, and c is subject 
to less variation. 

The inverse problem, to find the change in depth when l is 
given, cannot be directly solved by this formula, because the areas 
are functions of the depths. When d 2 — d\ is small compared with 
either d\ or d 2 , it is allowable to regard d 2 as equal to d\ when they 
are to be added or multiplied together. Hence 

i_i_ _ a 2 — a 2 _ d 2 — d\ _ {d 2 -\-d\) (d 2 — d\) _ 2(d 2 — di) 

a 2 a 2 a 2 a 2 b 2 d 2 d 2 b 2 d\ b 2 d\ 


also making a equal to a v and r equal to d\ in the last formula, 
and solving for d 2 — d\, there is found 


d 2 — d] _ i — q 2 /c 2 b 2 d* 
l i — q 2 /gb 2 di 


( 137 )* 


from which the change in depth can be computed when all the 
other quantities are given. 

Fig. 137 is drawn for the case of depth increasing down¬ 
stream, but the reasoning is general and the formulas apply 
equally well when the depth decreases with the fall of the stream. 
In the latter case the point A 2 is below C, and d 2 — d x will be 
negative. As an example, let it be required to determine the 
decrease in depth in a rectangular conduit 5 feet wide and 333 feet 
long, which is laid with its bottom level, the depth of water at 
the entrance being maintained at 2 feet, and the quantity sup¬ 
plied being 20 cubic feet per second. Here / = 333 ,b = 5, di = 2, 
q = zo, and i = o. Taking c = 89, and substituting all values 
in the formula, there is found d 2 —-di = —0.09 feet; whence d 2 = 
1.91 feet, which is to be regarded as an approximate probable 
value. It is likely that values of d 2 - di computed in this 
manner are liable to an uncertainty of 15 or 20 percent, the longer 
the distance l the greater being the error of the formula. In 
strictness also c varies with depth, but errors from this cause 
are small when compared to those arising in ascertaining its 
value from the tables. 

Prob. 137 . Explain why formula ( 137)2 cannot be used for the above 
example when the slope i is 0.01. 









348 


Chap. 10. The Flow of Rivers . 


Art. 138 . The Surface Curve 

In the case of steady uniform flow, in the channel where the 
bed has a constant grade, the slope of the water surface is paral¬ 
lel to that of the bed, and the longitudinal profile of the water 
surface is a straight line. In steady non-uniform flow', how r ever, 
the slope of the water surface continually varies, and the longi¬ 
tudinal profile is a curve whose nature is now to be investigated. 
As in the last article, the width of the channel will be taken as 
constant, its cross-section will be regarded as rectangular, and 
it will be assumed that the stream is wide compared to its depth, 
so that the wetted perimeter may be taken as equal to the width 
and the hydraulic radius equal to the mean depth (Art. 112 ). 
These assumptions are closely fulfilled in many canals and rivers. 

The last formula of the preceding article is rigidly exact if 
the sections d\ and a? are consecutive, so that / becomes 81 and 
d 2 — di becomes 8 d. Making these changes, 

« _ i - g-i cW 

81 

in which d is the depth of the water at the place considered. This 
is the general differential equation of the surface curve, l being 
measured parallel to the bed BB , and d upward, while the angle 
whose tangent is the derivative 8 d/8l is also measured from BB. 

To discuss this curve, let CC be the water surface if the slope 
were uniform, and let D be the depth of the water in the wdde 

A - --f-^_ A. 


Fig. 138a. Fig. 1386. 

rectangular channel. The slope 5 of the water surface is here 
equal to the slope i of the bed of the channel, and from the Chezy 
formula ( 113 ), 

q — av = cbD Vri = cbD V Di 











The Surface Curve. Art. 138 


349 


This value of q, inserted in the differential equation of the sur¬ 
face curve, reduces it to the form, 

8d = . i -(D/d) 3 

i - — (D/d) 3 ( 138 ) 2 

g 

in which d and l are the only variables, the former being the ordi¬ 
nate and the latter the abscissa, measured parallel to the bed BB , 
of any point of the surface curve. The derivative 8 d/8l is the 
tangent of the angle which the tangent at any point of the sur¬ 
face curve makes with the bed BB or the surface CC. 


First, suppose that D is less than d, as in Fig. 138 a, where 
A A is the surface curve under the non-uniform flow, and CC is 
the line which the surface would take in case of uniform flow. 
The numerator of ( 138)2 is then positive, and the denominator 
is also positive, since i is very small. Hence 8 d is positive, and 
it increases with d in the direction of the flow; going up-stream 
it decreases with d, and the surface curve becomes tangent to CC 
when d = D. This form of curve is that usually produced above 
a dam ; it is called the “ backwater curve,” and will be discussed 
in detail in Art. 140 . 


Second, let d be less than D, as in Fig. 1386 . The numerator 
is then negative and the denominator positive; 8 d is accordingly 
negative and AA is concave to the bed BB, whereas in the former 
case it was convex. This form of surface curve is produced when 
a sudden fall occurs in the stream below the point considered; 
it is called the “drop-down curve” and is discussed in Art. 141 . 

Formula ( 138 ) i may also be put into another form by substi¬ 
tuting for q its value bdv, where v is the mean velocity in the cross- 
section whose depth is d. It thus becomes 


8d _ g_ v 2 — c 2 di 
81 c 2 v 2 — gd 


( 138 ), 


and by its discussion the same conclusions are derived as before. 
When v is equal to c Vjf, the inclination 8 d/8l becomes zero, and 
the slope of the water surface is parallel to the bed of the stream. 
When v is less than c Vdf, the numerator is negative, and if the 




350 


Chap. 10. The Flow of Rivers 


denominator is also negative, the case of Fig. 138 a results. When 
v is greater than c Vdi and the denominator is negative, the case of 
Fig. 1385 obtains. When v equals Vgd, the value of M/U is 
infinity and the water surface stands normal to the bed of the 
stream; this remarkable case can actually occur in two ways, 
and they will be discussed in Art. 139 . 

Prob. 138 . Let the velocity of the stream be 20 feet per second, the 
value of c be 80, and the slope be 1 on 2000. Compute values of 8 d/Sl 
for depth of 12.2, 12.3, 12.4, 12.5, and 12.6 feet; then draw the surface curve. 


Art. 139 . The Jump and the Bore 



Fig. 139a. 


A very curious phenomenon which sometimes occurs in shallow 
channels is that of the so-called “jump,” as shown in Fig. 139 a. 

This happens when the denomi¬ 
nator in ( 138)3 is zero ; then M/M 
is infinite, and the water surface 
stands normal to the bed. Plac¬ 
ing that denominator equal to 
zero, there is found v 2 = gd. Now 
by further consideration it will appear that the varying denomi¬ 
nator in passing through zero changes its sign. Above the jump 
where the depth is d\ the velocity is slightly greater than V gdi, 
and below it is less than Vgd 2 . The conditions for the occurrence 
of the jump are that an obstruction should be in the stream below, 
that the slope i should not be small, and that the velocity Vi 
should be greater than \gd\. To find the necessary slope, the 
algebraic conditions are 

z>i = c \d\i and > Vgdi whence i>g/ c 2 


and accordingly the jump cannot occur when i is less than g/c 2 . 
For an unplaned planked trough c may be taken at about 100; 
hence the slope for this must be equal to or greater than 0.00322. 

To determine the height of the jump, let d 2 — d\ be represented 
by j. It is then to be observed that the lost velocity-head is 
(fli 2 — v 2 )/2g, and that this is lost in two ways, first by the 
impact due to the expansion of section (Art. 76 ), and second by 
the uplifting of the whole quantity of water through the height 


















The Jump and the Bore. Art. 139 


351 


l(do — d\), loss in friction between d\ and d 2 being neglected. 

Hence 2 2 ( \2 

vi ~ V2 = (V1-V2) | .7 

2 g 2g 2 


Inserting in this the value of v 2 , found from the relation 
z>2(di ~hj) = Vidi, and solving for 7, gives 


j=-d 1 + 2 . L ^ ( 139 ) 

\ 2 g 

The following is a comparison of heights of the jump computed 
by this formula and the observed values in four experiments made 
by Bidone, the depths being in feet: 


Depth rf. 

Velocity d. 

Observed j 

Computed j 

0.149 

4-59 

O.274 

0.290 

0.154 

4-47 

0.267 

0.283 

0.208 

5-59 

0.305 

0428 

0.246 

6.28 

0-493 

0.531 


The agreement is very fair, the computed values being all slightly 
greater than the observed, which should be the case, because the 
reasoning omits the frictional resistances between the points 
where d\ and d 2 are measured. Experiments made at Lehigh 
University, under velocities ranging from 2.2 to 6.2 feet per 
second, show also a good agreement between computed and ob¬ 
served value.* The depths in these experiments were less than 
in those of Bidone, but higher relative jumps were obtained. 
For instance, for v\ = 4.33 feet per second and d\ = 0.039 feet, 
the observed value of j was 0.166 feet, whereas the value com¬ 
puted from the above formula is 0.173 feet; here the jump is 
more than four times the depth d\, while it is usually less than 
twice d\ in the above records from Bidone. 

Another remarkable phenomenon is that of the so-called 
“bore,” where a tidal wave moves up a river with a vertical front. 
It is also seen when a large body of water moves down a canon 
after a heavy rainfall, or when a reservoir bursts and allows a 
large discharge to suddenly escape down a narrow valley. In the 
great flood of 1889 at Johnstown, Pa., such a vertical wall of water, 


* Engineering News, 1895, vol. 34, p. 28. 





352 


Chap. 10. The Flow of Rivers 


variously estimated at from io to 30 feet in height, was seen to 
move down the valley, carrying on its front brush and logs mingled 
with spray and foam.* In 41 minutes it traveled a distance of 

13 miles dow r n the descent of 380 feet. 
The velocity was hence about 28 feet 
per second. 

Fig. 1395 show r s the form of surface 
curve for this case, and by reference to 
( 138)3 it is seen that hd/hl must be negative and that it has 
the value 00 at the vertical front. The conditions for the occur¬ 
rence of the bore then are 

v='VJd and v>c vdi wdience i<g/c 2 



WMmmmmmmm 

Fig. 1396 . 


For the Johnstown flood, taking v as 28 feet per second, the value 
of d found from this equation is 24 feet; it w r as probably greater 
than this in the upper part of the valley and less in the low r er 
part. Since the value of i is about 1/180, it follows that c must 
have been less than 76. The conditions here established show 
that the flood bore will occur wdien the velocity becomes equal 
to Vg5, provided c is less than Vg/i. It appears, therefore, that 
roughness of surface is an essential condition for the formation 
of the bore in a steep valley. 

The bore can also occur in a canal with horizontal bed w r hen a lock 
breaks above an empty level reach, provided v becomes equal to ~\/gd. 
No case of this kind appears to be on record, and there seems to be no 
way of ascertaining w'hether the actual velocity will reach the limit 
'Vgd. If the bore occurs and the depth of the vertical wall be d 2 , its 
distance from a point where the depth is d x is found from ( 139) 2 by 
inserting in it the value of g corresponding to the critical velocity v. 
Thus may be shown that for c = 80 and d± = jd x the length l is 275^. 

The tidal bore, which occurs in many large rivers when the tide 
flows in at their mouths, obeys similar law r s. Here the slope i may be 
taken as zero, w r hile c is probably very large, so that roughness of sur¬ 
face is not an essential condition. The great bore at Hangchow, 
China, which occurs twice a year, is said to travel up the river at a rate 
of from 10 to 13 miles per hour, the height of the vertical front being 


Transactions American Society of Civil Engineers, 1889, vol. 21, p. 564. 









The Backwater Curve. Art. 140 


353 


from io to 20 feet.* From v = VgA, the velocity corresponding to a 
depth of io feet is 12.6 miles per hour, while that corresponding to a 
depth of 20 feet is 17 miles per hour, so that the statements have a fair 
agreement with the theoretical law. This investigation indicates that 
the velocity of the tidal bore depends mainly upon the depth of the 
tidal wave above the river surface, but it may be noted that other 
discussionsf regard the depth of the river itself as an element of impor¬ 
tance, and Art. 191 considers this with respect to common nvaves. 

Prob. 139 . When the height of the jump is three times the depth di, 
show that the velocity v\ must be 2->J2gd\. Also show that 0.414Gb is the 
minimum height of a jump. 


Art. 140 . The Backwater Curve 


When a dam is built across a channel the water surface is 
raised for a long distance up-stream. This is a fruitful source 
of contention, and accordingly many attempts have been made 
to discuss it theoretically, in order to be able to compute the 
probable increase in depth at various distances back from a pro¬ 
posed dam. None of these can be said to have been successful 
except for the simple case where the slope of the bed of the channel 
is constant and its cross-section such that the width may be re¬ 
garded as uniform and the hydraulic radius be taken as equal to 
the depth. These conditions are closely fulfilled for some streams, 
and an approximate solution may be made by the formula ( 137 ) 2 . 
It is desirable, however, to obtain an exact equation of the sur¬ 
face curve. 


For this purpose take the differential equation of the surface curve 
given in ( 138 ) 2 , and let the independent variable d/D be represented 
by x. Then it may be put into the more convenient form 


U_D( 1 . i-cn-M 

8 x i \ x 3 — 1 / 


( 140 )! 


in which l is the abscissa and Dx the ordinate of any point of the curve. 
The general integral of this is 


l = Hi - d(- - — 

i \i g 


i loe 


x 2 4 “ x ~b 1 




(x — i) 2 V3 


-- arc cot 


2 x 




V 3 / 


+ C 


* Skidmore’s China, the Long-lived Empire (New York), 1900, p. 294. 
f G. H. Darwin, The Tides, p. 65 ; Century Magazine, vol. 34, p. 903. 








354 


Chap. 10. The Flow of Rivers 


which is the equation of the surface curve, C being the constant of 
integration. To use this let the logarithmic and circular function 
in the second parenthesis of the second member be designated by 
cj>(x) or <f>(d/D)j namely, 


+(.*) = *WD) = l log. * 2 ( 


V3 


arc cot 


2 X + I 

V3 


Then the above value of / may be written 



Now let d 2 be the depth at the dam and let / be measured up-stream 
from that point to a section where the depth is d x . Then, taking the 
integral between these limits the constant C disappears, and 



which is the practical formula for use. In like manner d 2 may repre¬ 
sent a depth at any given section and d\ any depth at the distance l 

up the stream. 

When d = D, the depth of 
the backwater becomes equal to 
that of the previous uniform 
flow, # is unity, and hence l is 
infinity. The slope CC of uni¬ 
form flow is therefore an asymptote to the backwater curve. Accord¬ 
ingly the depth d 1 is always greater than /}, although practically 
the difference may be very small for a long distance l. 



In the investigation of backwater problems by the above formula 
there are two cases: first, d 2 and d 1 may be given and l is to be found ; 
and second, l and one of the depths are given and the other depth is 
to be found. To solve these problems the values of the backwater 
function $(d/D) computed by Bresse are given in Table 140 .* The 
argument of the table is D/d, which, being always less than unity, 
is more convenient for tabular purposes than d/D, since the values of 
the latter range from i to oo . By the help of Table 140 practical 
problems may be discussed and the following examples will illustrate 
the method of procedure. 


* Bresse’s Mecanique appliques (Paris, 1868), vol. 2, p. 556. 












The Backwater Curve. Art. 140 


355 


Table 140 . Values of the Backwater Function 


D 

d 

O 

D 

d 

bT»- 

D 

d 

O 

D 

d 

♦(f) 

I. 

00 

0.954 

0.9073 

0.845 

0.5037 

O.61 

0.2058 

0.999 

2.1834 

•952 

.8931 

.840 

•4932 

.60 

.1980 

.998 

I -9523 

•950 

•8795 

.835 

.4831 

•59 

.1905 

•997 

1.8172 

•948 

.8665 

.830 

•4733 

.58 

.1832 

.996 

1.7213 

.946 

•8539 

.825 

•4637 

•57 

.1761 

•995 

1.6469 

•944 

.8418 

.820 

•4544 

•56 

.1692 

•994 

1.5861 

.942 

.8301 

.815 

•4454 

•55 

.1625 

•993 

I -5348 

•940 

.8188 

.810 

•4367 

•54 

.1560 

.992 

1.4902 

.938 

.8079 

.805 

.4281 

•53 

.1497 

.991 

1.4510 

•936 

• 7973 

.800 

.4198 

•52 

•1435 

.990 

I- 4 I 59 

•934 

.7871 

•795 

.4117 

• 5 i 

.1376 

.989 

1.3841 

•932 

.7772 

.790 

•4039 

•50 

.1318 

.988 

I- 355 I 

•930 

•7675 

.785 

.3962 

•49 

.1262 

.987 

1.3284 

.928 

.7581 

.780 

.3886 

.48 

.1207 

.986 

1-3037 

.926 

.7490 

•775 

•3813 

•47 

•ii 54 

•985 

1.2807 

.924 

• 7401 

•770 

•3741 

.46 

.1102 

.984 

1.2592 

.922 

•7315 

•765 

.3671 

•45 

.1052 

•983 

1.2390 

.920 

•7231 

.760 

.3603 

•44 

.1003 

.982 

1.2199 

.918 

.7149 

•755 

•3536 

•43 

•0995 

.981 

1.2019 

.916 

.7069 

•750 

•3470 

.42 

.0909 

.980 

1.1848 

.914 

.6990 

•745 

.3406 

.41 

.0865 

•979 

1.1686 

.912 

.6914 

•740 

•3343 

.40 

.0821 

•978 

i-i 53 i 

.910 

.6839 

•735 

.3282 

•39 

•0779 

•977 

1-1383 

.908 

.6766 

• 730 

.3221 

.38 

.0738 

.976 

1.1241 

.906 

.6695 

•725 

•3 i 62 

•37 

.0699 

•975 

1.1105 

.904 

.6625 

.720 

•3 io 4 

•36 

.0660 

•974 

1.0974 

.902 

•6556 

• 7 i 5 

.3047 

•35 

.0623 

•973 

1.0848 

.900 

.6489 

.710 

.2991 

•34 

•0587 

.972 

1.0727 

•895 

.6327 

.705 

•2937 

•33 

•0553 

.971 

1.0610 

.890 

.6173 

• 70 

.2883 

•32 

.0519 

.970 

1.0497 

.885 

.6025 

.69 

.2778 

•30 

•0455 

.968 

1.0282 

.880 

.5884 

.68 

.2677 

.28 

•0395 

.966 

1.0080 

•875 

•5749 

.67 

.2580 

•25 

.0314 

.964 

0.9890 

.870 

.5619 

.66 

.2486 

.20 

.0201 

.962 

.9709 

.865 

•5494 

•65 

•2395 

•15 

.0113 

.960 

•9539 

.860 

•5374 

.64 

.2306 

.10 

.0050 

•958 

•9376 

.855 

•5258 

•63 

.2221 

•05 

.0015 

•956 

.9221 

.850 

.5146 

.62 

.2138 

.00 

.0000 






















356 


Chap. 10. The Flow of Rivers 


A stream of 5 feet depth is to be dammed so that the water shall 
be 10 feet deep a short distance up-stream from the dam. The uni¬ 
form slope of its bed and surface is 0.000189, or a little less than one 
foot per mile, and its channel is such that the coefficient c is 65- It 
is required to find at what distance up-stream the depth of water is 
6 feet. Here D = 5, d 2 = 10, d x = 6 feet, i/i = 5291, and c ~/g = 
131. Now D/d 2 = 0.5, for which the table gives <j>(djD) = 0.1318, 
and D/d x = 0.833, for which the table gives <P(d x /D) = 0.4792. 
These values inserted in ( 140) 2 give 

l = 5291(10 - 6) + 5(5291 - 131X0.4792 - 0.1318) 

from which l = 30 125 feet = 5.70 miles. In this case the water is 
raised one foot at a distance 5.7 miles up-stream from the dam. 

The inverse problem, to compute d 2 or d x , when one of these and 
/ are given, can only be solved by repeated trials by the help of Table 
140 . For example, let / = 30 125 feet, the other data as above, and 
let it be required to determine d 2 so that d x shall be only 5.2 feet, or 0.2 
greater than the original depth of 5 feet. Here D/d x = 0.962, for 
which the table gives d>{dJD) = 0.9709. Then ( 140) 2 becomes 

30 125 = 5291(3/2 - 5.2)4-25 800(0.9709 - d>(d 2 /D)] 
which is easily reduced to the simpler form 

32 590 = 5291 d 2 - 25 800 <t>(d 2 /D) 

Values of d 2 are now to be assumed until one is found that satisfies 
this equation. Let d 2 = 8 feet, then ( D/d 2 ) = 0.625 and, from the 
table, <t>(d 2 /D) = 0.2180; substituting these, the second member be¬ 
comes 36 700, which shows that the assumed value is too large. Again, 
take di = 7 feet, then D/d 2 = 0.714, for which <t>(d 2 /D) = 0.3047, 
whence the second member is 29 200, showing that 7 feet is too small. 
If d 2 = 7.4 feet, then D/d 2 = 0.675 an( I d>{d 2 /D) = 0.2629, and with 
these values the equation is nearly satisfied, but 7.4 is still too small. 
On trying 7.5 it is found to be too large. The value of d 2 hence lies 
between 7.4 and 7.5 feet, which is as close a solution as will generally 
be required. The height of dam required to maintain this depth may 
now be computed from Art. 136 . 

If the slope, width, or depth of the stream changes materially, 
the above method, in which the distance l is measured from the dam 
as an origin, cannot be used. In such cases the stream should be di- 


The Backwater Curve. Art. 140 


357 


vided into reaches, for each of which the slope, width, and depth can 
be regarded as constant. The formula can then be used for the first 
reach and the depth of its upper section be determined; then the ap¬ 
plication can be made to the next reach, and so on in order. For com¬ 
mon rivers and for shallow canals it will probably be a good plan to 
determine D by actual measurement of the area and wetted perimeter 
of the cross-section, the hydraulic radius computed from these being 
taken as the value of D. Strictly speaking, the coefficient c varies with 
the slope and with D, and its values may be found by Kutter ’s for¬ 
mula, if it be thought worth the while. Even if this be done, the results 
of the computations must be regarded as liable to considerable un¬ 
certainty. In computing depths for given lengths an uncertainty of 
io percent or more in the value of d^—di should be expected. 


The following method of computation is readily applicable to 
cases of backwater and gives results which are often sufficiently 
satisfactory. The distance l between two sections does not ap¬ 
pear in the formulas, but it is essential that this distance shall 
be small enough so that the water surface between them may be 
regarded as a straight line. In some streams the distance apart 
of sections may be as high as 1000 feet, in others smaller. Let 
Fig. 1405 represent the 
case of a stream where 
an obstruction, which 
is some distance down¬ 
stream from the sta¬ 
tion M, causes a rise of 
the original surface. 

At the several stations 



M, N, P, Q, R, etc., elevations of the original surface above 
a datum plane are taken. A cross-section of the stream is also 
made at each station, the levels being extended upward on the 
banks so that for any water level the area a and the wetted 
perimeter p may be ascertained from a drawing. At the first 
station M the elevation of the backwater is known, it being 
either assumed or computed from Art. 136 . The problem 
then is to determine the elevation of the backwater at each of 


the stations up-stream from M. 











358 


Chap. 10. The Flow of Rivers 


Fig. 140 c shows on a larger scale the profile between M and 
N and also the two cross-sections at M which are drawn from 
the given data. In this diagram the elevations of M h M 2 , and N\ 
are known, and it is required to find that of i¥ 2 . Let a\ and a 2 



denote the areas of the cross-section at M, the first for the original 
flow and the second for the backwater, and let pi and p 2 be the 
corresponding wetted perimeters. Let h\ be the known difference 
of the elevations of Mi and Ni, and h 2 the unknown difference 
of the elevations of M 2 and N 2 . Then the formula 

h = h i ^ ( 140) 3 

a 2 s pi 

determines h 2 , and accordingly the elevation of N 2 is known. This 
formula expresses the condition that the same quantity of water 
flows through the cross-sections a\ and a 2 , and it is deduced as 
follows. The mean discharges in these two sections are, from the 
Chezy formula, Cidi\r\Si and c 2 a 2 ^/r 2 s 2 . Equating these, re¬ 
placing r\ and r 2 by a\/pi and a 2 /p 2 , squaring, and making the 
coefficients Ci and c 2 equal, gives the equation Sid^/pi = s 2 d 2 3 /p 2 . 
Now Si = hil and s 2 = h>l where l is the distance between the two 
sections. Hence hidi/pi = k 2 d 2 3 /p 2 , from which the above for¬ 
mula ( 140) 3 at once results. 

As an example, take the case of four stations on Coal River, 
W.Va., data for the original water surface being as follows: 


Station 

= M 

N 

P 

Q 

R 

Elevation 

= 10.05 

n -53 

n -95 

13-44 

14.39 ft. 

Rise 

hi = 

1.48 

0.42 

1.49 

0.95 ft. 

Area 

01 = 3034 

3012 

3210 

2749 

2340 sq. ft. 

Perimeter pi — 255 

260 

280 

204 

192 ft. 


















The Backwater Curve. Art. 140 


359 


and let it be required to find the elevations of the backwater sur¬ 
face when an obstruction down-stream from M raises the water 
to elevation 12.05 at M 2 . Drawing the water level in the cross- 
section at M, there are found a 2 = 3533 square feet and p 2 = 260 


feet. Then 



1.48 


3034 3 X 260 
3533 s X 255 


= 0.95 feet, 


and hence the elevation at N 2 is 12.05 + 0.95 = 13.00 feet. For 
this water-level the cross-section for station N gives 3390 square 
feet area and 264 feet wetted perimeter for the backwater condi¬ 
tion. Then the backwater rise at station P is 


7 50I2 3 X 280 , 

h 2 = 0.42 52 -■ - - = 0.30 feet, 

3390 3 X 264 

which gives 13.30 feet for the elevation of the backwater surface 
at P. The results for the five stations are arranged as follows, 
the last line showing the required elevations of the backwater 
surface: 


Station 

= M 

N 

P 

Q 

R 

Area 

02= 3533 

3390 

358 o 

2940 

2492 sq. ft. 

Perimeter p 2 = 260 

264 

286 

209 

197 ft. 

Rise 

h 2 = 

o -95 

0.30 

1.10 

0.80 ft. 

Elevation 

= 12.05 

13.00 

13-30 

14.40 

15.20 ft. 


While there are several assumptions and limitations in this 
method, it does not appear that they introduce more error than 
that which obtains when the formula ( 140) 2 is applied to a stream 
of irregular section. By the exercise of much judgment in select¬ 
ing the stations, and by taking the data for a cross-section as 
the mean of several on both sides of a station, it is believed that 
the method can be used with much confidence in all cases where 
extreme conditions do not obtain. If the Chezy coefficients at a 
station can be found, then the formula ( 140)3 may be written in 
the more exact form 

h 2 = h\ Cidip 2 /c 2 a 2 p\ ( 140)4 

Prob. 140 . A stream, having a cross-section of 2400 square feet and 
a wetted perimeter of 300 feet, has a uniform slope of 2.07 feet per mile, and 
its channel is such that c = 70. It is proposed to build a dam to raise the 
water 6 feet above the former level, without increasing the width. Compute 
the rise of the backwater at a distance of one mile up-stream. 




360 


Chap. 10. The Flow of Rivers 


Art 141 . The Drop-down Surface Curve 


When a sudden fall occurs in a stream, the water surface for a long 
distance above it is concave to the bed, as seen in Fig. 1386 or in Fig. 

141 . This case also occurs when 
the entire discharge of a canal is 
allowed to flow out through a fore¬ 
bay F to supply a water-power 
plant. Let D be the original uni¬ 
form depth of water having its 
surface parallel to the bed, the 
slope of both being i. Let di and 
d 2 be two of the depths after the steady non-uniform flow has been 
established by letting water out at F, and let d l be greater than d 2 , the 
distance between them being l. The investigation of the last article 
applies in all respects to this form of surface curve, and 



Fig. 141 . 


l = - 


d\ — d< 2 , 




( 141 ) 


is the equation for practical use, in which c is the coefficient in the 
Chezy formula v = c^/rs, and g is the acceleration of gravity. Table 
140 cannot, however, be used for this case because d/D in that table 
is greater than unity, while here it is less than unity. 

The function d>(d/D) with values of d/D less than unity is here 
called the “drop-down function,” in order to distinguish it from the 
backwater function of the last article, although the algebraic expression 
for the two functions is the same. Table 141 , due also to Bresse, 
gives values of this drop-down function for values of the argument 
d/D, ranging from o to i, and by its use approximate solutions of prac¬ 
tical problems can be made. For example, take a canal io feet deep, 
having a coefficient c equal to 8o, and let the slope of its bed be i/5000 
and its surface slope be the same when the water is in uniform flow. 
Here D = 10 feet, c 2 /g = 200, and i/i = 5000. Then 


l = — 5000 (d\ — d 2) + 48 000 


XzO-nl 


Now suppose that a break occurs in the bank of the canal out of which 
rushes more water than that delivered in normal flow when the depth 
is 10 feet, and let it be required to find the distance between two points 
where the depths of water are 8 and 7 feet. Here d\/D = 0.8, for which 














361 


The Drop-down Surface Curve. Art. 141 


Table 141 . Values of the Drop-down Function 


d 

« / d \ 

d 

♦(5) 

D 

*(d) 

D 

I. 

CO 

0.954 

0.8916 

O.999 

2.1831 

•952 

.8767 

.998 

I-95I7 

•950 

.8624 

•997 

1.8162 

.948 

.8487 

.996 

1.7206 

.946 

•8354 

•995 

I.6452 

•944 

.8226 

•994 

1.5841 

•942 

.8102 

•993 

1-5324 

•940 

Cl 

OO 

O 

.992 

I.4876 

•938 

.7866 

.991 

I.4486 

•936 

•7753 

.990 

I.4125 

•934 

.7643 

.989 

1.3804 

•932 

.7537 

.988 

I-35H 

•930 

.7433 

.987 

1.3241 

.928 

•7332 

.986 

1.2990 

.926 

.7234 

•985 

1-2757 

.924 

00 

co 

w 

.984 

1.2538 

.922 

.7045 

.983 

1.2323 

.920 

•6953 

.982 

1.2139 

.918 

.6864 

.981 

I-I955 

.916 

.6776 

.980 

1.1781 

•9H 

.6691 

•979 

1.1615 

.912 

.6607 

•978 

1.1457 

.910 

•6525 

•977 

1-1305 

.908 

.6445 

.976 

1.1160 

.906 

.6366 

•975 

1.1020 

•9°4 

.6289 

•974 

1.0886 

.902 

.6213 

•973 

1-0757 

.900 

.6138 

.972 

1.0632 

.895 

•5958 

.971 

1.0512 

.890 

•5785 

.970 

1.0396 

.885 

.5619 

.968 

1.0174 

.880 

.5459 

.966 

0.9965 

.875 

•5305 

.964 

.9767 

bo 

0 

•5156 

.962 

•958o 

.865 

•5° 12 

.960 

.9402 

.860 

•4872 

.958 

•9233 

.855 

•4737 

•956 

.9071 

.850 

.4605 


d 

D 

*(d) 

d 

D 

♦(5) 

0.845 

0.4478 

O.61 

0.0454 

.840 

•4353 

.60 

•0325 

•835 

•4232 

•59 

.0199 

.830 

.4114 

.58 

+ 0.0074 

.825 

.3988 

•57 

— 0.0050 

.820 

.3886 

•56 

— .0172 

.815 

•3776 

•55 

- -0293 

.810 

.3668 

•54 

— .0412 

.805 

.3562 

•53 

- .0530 

.800 

•3459 

•52 

- .0647 

•795 

•3357 

•5i 

- .0763 

•790 

•3258 

•50 

— .0878 

.785 

• 3 i6 ° 

•49 

- .0991 

.780 

.3064 

.48 

— .1104 

•775 

.2970 

•47 

— .1216 

•770 

.2877 

.46 

- .1327 

•765 

.2785 

•45 

- .1438 

.760 

.2696 

•44 

- -1547 

• 755 

.2607 

•43 

- .1656 

•750 

.2520 

.42 

- -1765 

• 745 

• 2434 

.41 

— .1872 

• 740 

•2350 

.40 

— .1980 

•735 

.2260 

•39 

— .2086 

•730 

.2184 

•38 

— .2192 

•725 

.2102 

•37 

— .2298 

.720 

.2022 

•36 

- .2403 

.715 

•1943 

•35 

— .2508 

.710 

.1864 

•34 

— .2612 

•705 

.1787 

•33 

— .2716 

•70 

.1711 

•3 2 

— .2819 

.69 

.1560 

•30 

- -3025 

.68 

• 1413 

.28 

- -3230 

.67 

.1268 

•25 

- -3536 

.66 

.1127 

.20 

— .4042 

•65 

.0987 

•15 

- -4544 

.64 

.0851 

.10 

- .5046 

•63 

.0716 

•05 

- .5546 

.62 

.0584 

.00 

— .6046 













































362 


Chap. 10. The Flow of Rivers 


4 >(di/D) = 0.3459, and d^/D = 0.7, for which d>{di/D) = 0.1711. In¬ 
serting these values in the equation, there is found l = 7890 feet. 

In this case there is a certain limiting depth below which the above 
formula is not valid. This limit is the value of x for which SI 
becomes zero or the value of x where the surface curve is vertical and 
the bore occurs (Art. 139 ). From ( 140 )i this happens when 

r 3 = c H/g or d = D{c 2 i/gY 

and for the above example this limiting depth is found to be 3.4 feet. 
Near this limit, however, the velocity becomes large, so that there is 
much uncertainty regarding the value of the coefficient c. 

When a given discharge per second is taken out of a forebay at the 
end of a canal having its bed on a slope i, the above formula must be 
modified. Let q be the discharge and let D\ be the depth at a section 
where the slope is 5, then q equals cbD\ V D y s. If this value of q be sub¬ 
stituted in the equation ( 138 ) 1 and then the same reasoning be followed 
as at the beginning of Art. 140 , it will be found that formula ( 141 ) 
will apply to this case if D\(s/i )* be used instead of D. For example, 
let q = 3000 cubic feet per second, D\ = 10 feet, i — 1/10 000, c = 80, 
and the width b = 100 feet. Then 

s = q 2 /c 2 b 2 D y 3 = 1/7100 D = D\(s/iY = 11.2 feet. 

Now if it be required to find the distance between two points where the 
depths of water are 10 and 9 feet, formula ( 141 ) can be directly applied, 
and accordingly there is found, by the help of Table 141 , 

l = — io 000(10 — 9) + 109 800(0.578 — 0.355) = 14 400 feet, 

and hence a forebay admitting the given discharge will not draw down 
the water to a depth less than 9 feet if it be located 14 400 feet down¬ 
stream from the section where the mean depth is 10 feet. 

Navigation canals are often built with the bed horizontal between 
locks, and here i = o. The above formula cannot be applied to this 
case because the differential equation ( 138)2 vanishes when i is zero. 
To discuss it, equation ( 138 )i must be resumed, and, inverting the same, 

SI _ cW . c 2 
&d q 2 g 

The integration of this between the limits d\ and dz gives 

l = ~ W - <V) - - (rfi - d,) 

4 ?- g 


(HI), 





The Drop-down Surface Curve. Art. 141 


363 


from which l may be computed when q is known. As an example, 
take a rectangular trough for which q = 20 cubic feet per second, 
b = 5 feet, c = 89, and let d\ — 2.00 feet and d 2 = 1.91 feet. Then 
from the formula l is found to be 317 feet. This is the reverse of the 
example at the end of Art. 137 , where l was given as 333 feet, so that 
the agreement is very good. 

To compare a canal having a level bed with the one previously 
considered, the same data will be used, namely, di = 10 feet, d 2 = 9 
feet, b = 100 feet, c = 80, and q = 3000 cubic feet per second. Then 
from ( 141)2 there is found 

l = i.778(io 4 — 9 4 ) — 200(10 — 9) = 5920 feet, 


and accordingly the water level is drawn down in one-third of the dis¬ 
tance of that of the previous case. The quantity of water that can be 
obtained from a navigation canal is always less than from one having 
a sloping bed, and it has frequently happened, when such a canal is 
abandoned for navigation purposes and is used to furnish water for 
power or for a public supply, that the quantity delivered is very much 
smaller than was expected. 

The method of computation explained at the end of Art. 140 
may be used also to determine the drop-down curve. Referring 
to Fig. 1406 the upper curve will be the original one and the lower 
one that which is obtained by computation. The formula ( 140) 3 
is to be used by taking hi, a x , pi for the upper curve and h 2 , a 2 , p 2 
for the lower one. For example, let the data for a station on 
the upper original curve be a x = 600 square feet and pi = 80 
feet, a 2 = 480 square feet and p 2 = 66 feet. Let the elevations 
of two points on the upper curve be 18.26 and 16.68 feet so that 
hi = 1.58 feet, then the fall in the lower curve is 


h 2 


1.58 


600 3 X 66 
480 3 X 80 


= 2.57 feet, 


and hence when the elevation of the first station on the lower 
curve is 16.26 feet, the probable elevation of the second station 
on that curve is 13.69 feet. The fall 2.57 feet is here probably 
liable to a considerable error, since the application of (141 )i to 
these data gives a much smaller result for h 2 . Experiments 
are greatly needed in order to test the comparative value of 



364 


Chap. 10. The Flow of Rivers 


these two methods of computation, and these, on a small scale, 
might well be undertaken in the hydraulic laboratory of an engi¬ 
neering college. 

Prob. 141 a. A canal from a river to a power house is two miles long, 
its bed is on a slope of i/io ooo, and c is 70. When the water is in uniform 
flow, the depth D is 6.0 feet, and the discharge is 800 cubic feet per second. 
If there be a power house which takes 1000 cubic feet per second, find the 
probable depth of water at the entrance to its forebay. 

Prob. 1416 . Show that the last formula in Art. 135 , when reduced to. 
the metric system, becomes v = v' + 6.1 Vrs. 

Prob. 141 c. A stream 181 meters wide and 5 meters deep has a dis¬ 
charge of 1318 cubic meters per second. Find the height of backwater 
when the stream is contracted by piers and abutments to a width of 96 meters. 

Prob. 14 Id. Which has the greater discharge, a stream 1.2 meters deep 
and 20 meters wide on a slope of 3 meters per kilometer, or a stream 1.6 
meters deep and 26 meters wide on a slope of 2 meters per kilometer ? 

Prob. 141 c. A stream 2 meters deep is to be dammed so that water shall 
be 4 meters deep at the dam. Its slope is 0.0002 and its channel is such that 
the metric value of c is 39. Compute the distance to a section up-stream 
where the depth of water is 3.6 meters. 


Rainfall. Art. 142 


365 


CHAPTER 11 

WATER SUPPLY AND WATER POWER 
Art. 142 . Rainfall 

All the water that flows in a stream has at some previous time 
been precipitated in the form of rain or snow. The word “ rain¬ 
fall” means the total rain and melted snow, and it is usually 
measured in vertical inches of water. The annual rainfall is 
least in the frigid zone and greatest in the torrid zone; at the 
equator it is about ioo inches, at latitude 40° about 40 inches, 
and at latitude 6o° about 20 inches. There are, however, cer¬ 
tain places where the annual rainfall is as high as 500 inches, and 
others where no rain ever falls. In the United States the heaviest 
annual rainfall is near the Gulf of Mexico, where 60 inches is 
sometimes registered, and near Puget Sound, where 90 inches is 
not uncommon. In that large region, formerly 
called the Great American Desert, which lies be¬ 
tween the Rocky and Sierra Nevada mountains, 
the mean annual rainfall does not exceed 15 
inches, and in Nevada it is only about 7^ inches. 

The amount of rainfall in any locality depends 
upon the winds and upon the neighboring moun¬ 
tains and oceans. 

The standard type of rain gage used by the 
U. S. Weather Bureau has a diameter of 8 inches. 

The rain falling into the gage passes down through 
the funnel shown in Fig. 142 a and into the small 
cylinder A, the area of which is one-tenth that of 
the gage. One inch of rainfall therefore will give a 
depth of 10 inches in the cylinder A and small falls can thus be 
accurately measured. As the cylinder A fills it overflows into 








366 Chap. 11 . Water Supply and Water Power 

the body of the gage B, and when measured is simply poured 
into the cylinder A after the water it contains has been measured 
and poured out. These gages should be read each day in 
order that the loss due to evaporation may not become exces¬ 
sive and introduce material errors. Other forms of rain gages 
which record on a chart each one-hundredth of an inch of 
rainfall at the time when it falls are made. Such gages are 
of particular use in determining the rate of rainfall and the time 
of the fall rather than its total quantity. 

At any place the rainfall in a given year may vary consider¬ 
ably from the mean derived from the observations of several 
years. Thus, at Philadelphia, Pa., the mean annual rainfall is 
about 42 inches, but in 1890 it was 50.8 inches and in 1885 it was 
only 33.4 inches. Similarly at Denver, Col., the mean is about 
14 inches, but the extremes are about 20 and 9 inches. When 
a very low rainfall occurs, that of the year preceding or following 
is also apt to be low, and estimates for the water supply of towns 
must take into account this minimum annual rainfall. The 
distribution of rainfall throughout the year must also be con¬ 
sidered, and for this purpose the rainfall records of the given 
locality should be obtained from the publications of the U. S. 
Weather Bureau as well as from all other available sources 
and be carefully discussed. In making plans for a water supply 
it should be the aim to store a sufficient quantity so that an ample 
amount will be available at the end of the driest period which is 
likely to occur. In Table 142 are shown the average rainfalls 
at a number of places in the United States for the four seasons 
and for the year; in estimates for very wet years about 25 per¬ 
cent may be added to these values, while for very dry years about 
25 percent may be subtracted. 

As illustrating the variations from the mean rainfall which 
may be expected at any place the following example is given. 
The mean rainfall at Philadelphia is about 42 inches, and the 
following are some of the values for various years: 29.6 inches for 
1825, 30.2 inches for 1881, 61.3 inches for 1867, and 55.5 inches 
for 1840. 


Rainfall. Art. 142 


367 


Table 142 . Rainfall in the United States * 


City 

Length of 
Record. 
Years 

Rainfall in Inches 

Spring 

Summer 

Autumn 

Winter 

Annual 

Vicksburg . 

32 

15-9 

12.0 

10.3 

15.6 

53-8 

Charleston. 

33 

10.6 

20.1 

I2 -5 

10.2 

53-4 

Little Rock .... 

24 

14-5 

II .2 

10.5 

13-4 

49.6 

Portland. 

32 

10.7 

3-0 

11.9 

20.0 

45-6 

New York. 

33 

10.6 

12.3 

10.8 

11.1 

44.8 

Boston. 

3 i 

11.2 

10.5 

11.1 

10.9 

43-7 

Cairo. 

22 

11.4 

10.4 

9.1 

10.7 

41.6 

Cincinnati. 

33 

9.9 

10.9 

7-9 

9-7 

38.4 

Key West. 

33 

5-5 

12.6 

14-5 

5-3 

37-9 

Cleveland. 

33 

8-5 

10.2 

9.0 

7-9 

35-6 

Chicago. 

33 

8-7 

IO.I 

8.2 

6.4 

33-4 

Detroit. 

33 

7-9 

IO.I 

7.6 

6.6 

32.2 

Omaha. 

33 

8.8 

13-3 

6.4 

2.3 

30.8 

St. Paul. 

3 i 

7-4 

11.4 

7.0 

2.8 

28.6 

San Antonio .... 

18 

7-7 

8.4 

7.0 

5-3 

28.4 

San Francisco . . . 

32 

5-7 

0.2 

4-4 

12.2 

22.5 

Bismarck. 

29 

5-8 

8-3 

2.7 

2.0 

18.8 

Spokane . 

23 

4.1 

2.7 

4-7 

6.8 

18.3 

Salt Lake City . . . 

30 

5-9 

2.0 

3-8 

4.1 

15.8 

Los Angeles .... 

28 

i -7 

0.0 

5-6 

8.1 

15-4 

Santa Fe. 

30 

2.7 

6.2 

3-3 

2.0 

14.2 

Denver. 

3 i 

5-4 

4.4 

2.2 

i -7 

13-7 

Helena. 

24 

4.0 

3-9 

2.8 

2.6 

13-3 

Y uma. 

28 

0.4 

0.4 

0.6 

i -3 

2.7 


The annual rainfall at any locality seems to vary in cycles, 
but no law of such variation, if any there be, has yet been dis¬ 
covered. The manner of variation at Philadelphia and New 
York is shown in Fig. 1426 , the curves being obtained by plot¬ 
ting for each year a value for the rainfall which is one-third of 
the sum of the rainfalls for that year, the preceding year, and the 
following year. The curves are not drawn to exactly follow the 
plotted points, but are smoothed out in order to better illustrate 
the probable variations. 


* From Records of U. S. Weather Bureau to 1910. 

























368 Chap. 11 . Water Supply and Water Power 

The distribution of rainfall from place to place is also subject 
to many variations, some local and others general in their nature. 
Among them may be mentioned both the topography and the 



altitude of the country and their relation to the prevailing wind 
direction. The presence of large bodies of water in the neigh¬ 
borhood also has its influence. 

As examples of such variations in rainfall there may be mentioned 
the Esopus and Catskill watersheds in New York.* Their areas are 
nearly the same, they both drain into the Hudson River from the west, 
and their centers are not more than 25 miles apart, yet the rainfall on 
the former is about 20 percent greater than on the latter. As one other 
example there may be mentioned the rainfall at “ Number 4” in 
northern New York in the Western Adirondacks and Avon on the 
Genessee River 23 miles south of Lake Ontario. These two stations 
are but 145 miles apart, yet the average yearly rainfall at the former is 
50.4 inches, while at the latter it is only 27.0 inches. In determining the 
rainfall at any point or for any given area all available records must 
be examined and all other collateral evidence carefully analyzed, 
particularly in cases where estimates of the stream flow are to be based 
on estimates of the rainfall. 

Prob. 142 . Consult the‘‘ Instructions for Yoluntary Observers,” pub¬ 
lished by the United States Weather Bureau, and describe a method of 
determining the amount of rainfall contained in a given depth of snowfall. 
In making reports how much rainfall on the average is to be taken as 
representing a snowfall of 12 inches? 


* Monthly Weather Review, March, 1907. 
































Evaporation. Art. 143 


369 


Art. 143 . Evaporation 

After rain has fallen evaporation from both land and water 
surfaces at once begins and continues until all of the rainfall has 
passed off into the atmosphere, where it is condensed into clouds 
and again falls as rain, thus completing the cycle. Like rainfall 
the evaporation is to be measured in inches of depth. Various 
experiments on the evaporation from water surfaces have been 
made, and a number of the results which have been derived are 
shown in Table 143 a. 


Table 143 a. Monthly and Yearly Evaporation from 

Water Surfaqes 


Place 





Evaporation in Inches 





Jan. 

Feb. 

Mar. 

April 

May 

June 

July 

Aug. 

Sept. 

| Oct. 

Nov. 

Dec. 

Year 

Boston, 

Mass.* * * § 

0.96 

1.05 

1.70 

2.97 

4.46 

5-54 

5-98 

5-50 

4.12 

3.16 

2.25 

I - 5 I 

39.20 

Rochester, 

N.Y.f 

0.52 

o -54 

i -33 

2.62 

3-93 

4.94 

5-47 

5-30 

4-15 

3. 16 

i -45 

1.1 3 

34-54 

Emdrup, 
Denmark J 

0.70 

0.50 

0.90 

2.00 

3-70 

5 - 4 ° 

5.20 

4.40 

2.60 

1.30 

0.70 

0.50 

27.90 

Lee Bridge, 
England § 

o- 7 S 

0.60 

1.07 

2.10 

2-75 

3-14 

3-44 

2.85 

1.61 

1.06 

0.67 

0-57 

20.61 

GraniteReef, 
Arizona ^ 

4-25 

4.40 

5-25 

7.00 

9-50 

12.00 

12.75 

12.50 

11.00 

8.31 

6.56 

4.22 

97-74 

Birmingham, 
Ala A 

1.50 

1.50 

2.25 

4-45 

5 - 9 i 

7.28 

7-36 

7-34 

6.00 

4.00 

2.25 

1.50 

51-34 

Klamath, 
Oregon ^ 

0.50 | 

1.25 

3-57 

6.64 

7-15 

6.99 

8.01 

9.21 

6.13 

2.50 

1.00 

0.50 

53-45 


Evaporation from land surfaces is dependent on the character 
of the soil, on the extent and character of the forestation and cul¬ 
tivation, and in a considerable measure on the general steepness 
of the surface, for on this is dependent the time in which evapora¬ 
tion can act. In a steep country the rainfall rapidly runs into 

* Transactions, American Society of Civil Engineers, vol. 15. 

f Annual Reports, Rochester, N.Y., Board of Water Commissioners, 

t Hydrology, Beardmore, London, 1862. 

§ Proceedings, Institution Civil Engineers, vol. 45. 

Engineering News, June 16, 1910. 












































370 Chap. 11 . Water Supply and Water Power 

the streams, while in a flat country it passes off more slowly, and 
the amount of the evaporation is thus increased. 

Experiments on the evaporation from earth, from short 
grass and long grass surfaces have been made, and some results 
are shown in Table 1436 . 


Table 1436 . Monthly and Yearly Evaporation from Land 

Surfaces 


Place 






Evaporation 

in Inches 





Jan. 

Feb. 

Mar. 

April 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

Year 

Lancashire, 
England,* 
from earth 

0.64 

0-95 

i -59 

• 

2-59 

4.38 

3-84 

4.02 

3.16 

2.02 

1.28 

0.81 

0.47 

25-65 

Cumberland, 
England,! 
from earth 

0.95 

1.01 

1.77 

2.71 

4.11 

4-25 

4-13 

3-29 

2.96 

1.76 

1.25 

1.02 

29.21 

Emdrup, 
Denmark,* 
from short 

grass 

0.70 

0.80 

1.20 

2.60 

4.10 

5-50 

5.20 

4.70 

2.80 

1-30 

0.70 

0.50 

30.10 

Emdrup, 
Denmark,* 
from long 
grass 

0.90 

0.60 

1.40 

2.60 

4.70 

6.70 

9-30 

7.90 

5.20 

2.90 

1.30 

0.50 

44.00 

Rothamsted, 
England,! 
from earth 

o -45 

0.60 

0.88 

1-53 

1.69 

1.92 

2.26 

1-95 

2 .II 

1.70 

0.98 

0.61 

16.68 


The evaporation from any particular watershed is dependent 
on the tempeiature, the humidity, the altitude, the area of the 
watershed, and the area of the water surface on it. The evap¬ 
oration is dependent also on the wind velocity, the inclination 
or slope of the watershed, its geological character, its forest cover, 
and its state as regards cultivated areas. The total amount of 
evaporation is also dependent on the rainfall, and varies with it. 
The distribution of the rainfall throughout the year greatly in¬ 
fluences the evaporation; a heavy winter and a light summer 
rainfall will together show a small annual evaporation. 

* Hydrology, Beardmore, London. 

f Fanning, Treatise on Water Supply Engineering, New York, 1878. 

t Proceedings Institution Civil Engineers, vol. 105. 




































Evaporation. Art. 143 


371 


In the Atlantic States it may be said that the annual evapora¬ 
tion from land surfaces is about 45 percent and that from water 
surfaces about 60 percent of the annual rainfall, so that about 
one-half of the rainfall reaches the streams and may be utilized. 
In the arid regions west of the Rocky Mountains the percentages 
of evaporation are much higher, as indicated in Table 143 a. 

Many attempts to deduce a formula which will take account of 
the various factors which influence evaporation have been made but 
without definite success. The problem is a very complicated one. 
Vermeule has deduced the formula 

£=( i 5.5 + °. ] [6 7?)(o.o5 T — 1.48) 

where R is the annual rainfall and E the annual evaporation in inches, 
and T is the mean annual temperature in Fahrenheit degrees.* If 
T = 49 0 .6, this becomes E— 15.5 +0.16 R, which is a mean value for 
New Jersey and neighboring states; if T be 47 0 , the evaporation is 10 
percent less, and if T be 52 0 , it is 10 percent more, than this mean. 
The evaporation in different months varies greatly, the mean monthly 
temperature being the controlling factor. The following are average 
values given by Vermeule for the vicinity of New Jersey, where the 
mean annual temperature is 49°.6; r representing mean monthly rain¬ 
fall and e mean monthly evaporations in inches: 


Jan., 

e = 0.27 + o.ior 

July, 

e = 3.00 + o.3or 

Feb., 

e = 0.30 + 0.10 r 

Aug., 

e = 2.62 + o.25r 

March, 

e = 0.48 + o.ior 

Sept., 

e = 1.63 + 0.20 r 

April, 

e = 0.87 + o.ior 

Oct., 

e = 0.88 + o.i2r 

May, 

e = 1.87 + 0.20T 

Nov., 

e = 0.66 + o.ior 

June, 

e = 2.50 + o.25r 

Dec., 

e = 0.42 + o.ior 


To obtain the monthly evaporations for places of mean annual tem¬ 
perature T, the values found for e are to be multiplied by 0.05 T — 
1.48. Thus, if there be 8 inches of rain in July, e = 5.40 inches, and 
if the mean annual temperature be 56°, this is to be increased by 32 
percent. Vermeule’s formulas for evaporation were deduced from 
a consideration of the relation between the rainfall and the observed 
flows of a number of streams in the New England and Middle States. 
They take account of the effect of unequal distribution of the rainfall 

* U. S. Geological Survey of New Jersey (Trenton, 1894), vol. 3, p. 76. 


372 


Chap. 11 . Water Supply and Water Power 


throughout the year and give results which agree well with actual 
gagings if care be taken to determine a proper factor for each water¬ 
shed to which they are applied.* 

Like rainfall the evaporation varies greatly, even in regions not 
widely separated. In Art. 142 the difference in the rainfall on the 
Esopus and Schoharie watersheds in New York State was referred to. 
The evaporation on the Esopus will probably average about 15 inches 
per year, while on the Catskill it is not far from 19 inches, a differ¬ 
ence of over 20 percent in a distance of less than 30 miles. 

Experiments on evaporation are of interest and value, but the best 
results as to its amount are determined by taking the difference between 
the amount of the rainfall and the results of measured stream flows. 
In this manner all of the factors are taken account of and the most 
accurate results obtained. Experiments made by collecting the rain¬ 
fall in pans and measuring the depth of water from time to time are not 
highly reliable, since the size of the pan influences the results. It 
has been shown by the U. S. Department of Agriculture that the evap¬ 
oration from a pan 2 feet in diameter is about 75 percent, that from 
a pan 4 feet in diameter is about 50 percent, and that from a pan 6 
feet in diameter is about 30 percent greater than the evaporation 
from a large pond or lake.f 

Prob. 143. The rainfall on a watershed of 850 square miles is 44.8 
inches. Assuming a seasonal distribution as at New York (Table 142) 
compute the evaporation by Vermeule’s formula. 

Art. 144 . Ground Water and Runoff 

When the ground is frozen and the precipitation does not 
accumulate in the form of ice and snow, the runoff from a water¬ 
shed is closely equal to the rainfall minus the evaporation. If 
three inches of rain falls per month and one-third of this evaporates, 
the runoff will be nearly 2 cubic feet per second for each square 
mile of the watershed. The discharge due to a heavy rainfall 
occurring in a short period or to the melting of snow may be 
twenty or thirty times as great. A rainfall of 10 inches occurring 
in two days, if three-fourths of it is delivered at once to the 
streams, will give a flood discharge of about 100 cubic feet per 

* Monthly Weather Review, March, 1907. 
f American Civil Engineers’ Pocket Book, 1911, p. 1286. 


Ground Water and Runoff. Art. 144 


373 


second per square mile of watershed area. It is not usually 
necessary to consider these flood discharges in estimates for water 
supply and water power, except in order to take precautions 
against the damage they may cause. 

In Table 144 <z are shown some observed flood flows of various 
small and large streams in the United States. 

V 

Table 144 a. Observed Maximum Flood Flows * 


Stream and Place 

Watershed 

Area 

Square Miles 

Cubic Feet per 
Second per 
Square Mile 

Starch Factory Creek, New Hartford, N.Y. . . . 

3-4 

209 

Mad Brook, Sherburne, N.Y. 

5 -o 

262 

Mill Brook, Edmeston, N.Y. 

9.4 

241 

Sawkill, near mouth, N.J. 

35 -o 

229 

Rock Creek, Washington, D.C. 

77-5 

126 

Ramapo River, Mahwah, N.J. 

118.0 

105 

Esopus, Olive Bridge, N.Y. 

238.0 

no 

Great River, Westfield, Mass. 

350-0 

152 

Raritan River, Bound Brook, N.J. 

879 

59 

Mora River, La Cueva, N.M. 

159 

140 

Delaware River, Lamberttflle, N.J. 

6 500 

54 

Susquehanna River, Harrisburg, Pa. 

24 030 

19 


Data such as those in Table 144 a are of use in proportioning 
overflows and waste-weirs for reservoirs and in fixing on the length 
of overfall dams in rivers. Numerous formulas have been pro¬ 
posed, but data such as actual observations are to be preferred 
in making designs of this character. In each particular case all 
available information must be considered, including the traditions 
as to the past highest water, and then after making due allowance 
for all of the conditions which influence the rapidity of runoff 
from the watershed, a liberal factor of safety must be applied. 

Runoff may be defined as the difference between the rainfall 
and the evaporation if in the latter be included all of the water 
which fails to reach the streams. The runoff of a stream can 

* Taken largely from American Civil Engineers’ Pocket Book, New 
York, 1911. 




















374 


Chap. 11 . Water Supply and Water Power 


be determined by measuring the flow over a weir (Chap. 6) or 
by daily gage height readings in connection with a discharge curve 
which has been determined by gagings of the flow at various 
water stages (Art. 131 ). The runoff is usually expressed as a 
percentage of the rainfall, thus if F be the rainfall, E the evapora¬ 
tion, and R the runoff, all in inches, then R = F — E, and as a 
percentage of the rainfall the runoff is ioo(.F — E)/F. 

In Table 1446 are shown some observed values of the rainfall 
and runoff on a number of streams in the United States. 


Table 1446. Observed Rainfall and Runoff * 


Stream and Place 

Area of 
Watershed 
Square 
Miles 

Rainfall in 
Inches 

Ri 

Percent 
of Rainfall 

jnoff 

Cubic Feet 
per Second 
per Square 
Mile 

Sudbury, Boston, Mass. 

75-2 

45-77 

48.6 

I.64 

Connecticut, Hartford, Conn. . . . 

IO 234.O 

44.69 

56.5 

1.86 

Croton, Old Croton Dam, N.Y. . . 

338.0 

48.38 

50.8 

1.81 

Upper Hudson, Mechanicsville, N.Y. 

4 5 °°.o 

39-70 

59-0 

1.72 

Perkiomen, Philadelphia, Pa. . . . 

152.0 

47.98 

49.2 

1.74 

Potomac, Point of Rocks, Md. . . . 

9 650.0 

36.86 

38.6 

1.05 

Savannah, Augusta, Ga. 

7 294-0 

45-41 

48.9 

1.63 

Upper Mississippi, Pokegama Falls . 

3 265.0 

26.57 

18.4 

0.36 


The gagings which have been made and are being continued 
by the U. S. Geological Survey on many streams all over this 
country furnish a vast fund of information concerning the run¬ 
off of streams. The results of these gagings are published in the 
various Water Supply and Irrigation Papers of the Survey, and 
are to be consulted wherever questions involving the runoff of 
streams are being considered. 

During the spring the ground is filled with water which is 
slowly flowing toward the streams, and this ground water is the 
main source of the runoff from a watershed during the dry 
months. The velocity of flow of this ground water varies directly 
as the slope of its surface, for this velocity is so slow that no losses 

* From American Civil Engineers’ Pocket Book, New York, 1911. 

















Ground Water and Runoff. Art. 144 


375 


occur in impact (Art. 90 ). When the slope of the surface of the 
ground water becomes zero, the streams are dry if there be no 
rainfall. The discharge of a stream in a dry season hence depends 
upon the depth and slope of the ground water, and this in turn 
depends upon the previous rainfall, the topography of the country, 
and the character of the soil. 

While data regarding rainfall and evaporation will furnish 
valuable information regarding the mean annual flow of a stream, 
they will usually fail to indicate the mean discharge during differ¬ 
ent months. For this purpose the study of discharge curves 
and gage heights (Art. 134 ) is important, and if there be none 
for the stream in hand, it will be necessary to make a few gagings 
at different stages of water and to collect information regarding 
the lowest stages that have been observed in dry years. 

In irrigation work quantities of water are often estimated 
in terms of a convenient unit called the acre-foot, which is the 
quantity which will cover one acre to a depth of one foot, namely, 
43 560 cubic feet. The discharge of a stream is often stated in 
acre-feet per day. One acre-foot per day is 0.5042 cubic feet per 
second, or one cubic foot per second is 1.983 acre-feet per day. 
One acre-foot of water is 325851 U. S. gallons, and 1000000 
gallons is 3.0689 acre-feet. One inch of rainfall per month is, 
very closely, 0.9 cubic feet per second per square mile. 

In irrigation estimates the “duty” of water is to be regarded. 
This is defined as the number of acres that can be irrigated by a 
supply of one cubic foot per second, and it usually ranges from 
60 to 100 acres. An inverse measure of duty is the number of 
vertical inches of water required to irrigate any area, this usually 
ranging from 18 to 24 inches per year. The acre-foot is also fre¬ 
quently used in statements of duty of water. The methods of 
measuring the water by orifices and modules in terms of the 
miner’s inch unit have been explained in Art. 55 . 

The hydraulics of irrigation engineering differs in no respect from 
that of water supply and water power. Water is collected in reservoirs 
or obtained by damming a river, and it is led by a main canal to the 
area to be irrigated, and there it is distributed through smaller lateral 


376 Chap. 11 . Water Supply and Water Power 

canals to the fields. The smaller the canal or ditch, the steeper be¬ 
comes its slope, and in the final application to the crops the flow in 
the furrows is often normal to the contours of the surface. In a river 
system the brooks feed the creeks, and the creeks feed the river, the 
flow being from the smaller to the larger; in an artificial irrigation 
system, however, the flow is from the larger to the smaller channel. 

Seepage into the earth from an irrigation canal constantly goes 
on, unless its bed be puddled with clay or lined with concrete, and 
this loss of water is often very heavy. For new canals it is often as 
high as 50 percent of the water, but for old canals it may become 
lower than 10 percent. In making estimates for an irrigation supply 
it is hence necessary to take into account this seepage loss, and also 
to consider that due to evaporation. 

Prob. 144 . If all the rainfall that does not evaporate flows into the 
stream, find the runoff in cubic feet per second from a watershed of 1225 
square miles during a month when the rainfall is 3.6 inches, the mean annual 
temperature being 48°.5 Fahrenheit. Also for the temperature of 49°.5. 

Art. 145 . Estimates for Water Supply 

The consumption of water in American cities is, on the average, 
about 100 gallons per person per day, the large cities using more 
and the small ones less than this amount. The daily consump¬ 
tion in July and August is from 15 to 20 percent greater than the 
mean, owing to the use of water for sprinkling, while during 
January and February it is also greater than the mean in the 
colder localities, owing to the large amount that is allowed to run to 
waste in houses in order to prevent the freezing of the pipes. On 
Mondays, in small towns when every household is at work on 
the weekly washing, the consumption may be put at 50 percent 
higher than the mean for the week. Accordingly if the yearly 
mean be 100 gallons per person per day, the Monday consump¬ 
tion during very hot or very cold weather may be as high as 150 
gallons per person per day. When a large fire occurs, the hourly 
consumption for this purpose alone in a fire district of 10 000 
people may be at the rate of 175 gallons per person per day. In 
general the maximum available hourly supply should be from 
three to four times as great as the mean daily consumption. 


Estimates for Water Supply. Art. 145 377 

When water is to be pumped from a river directly into the 
pipes, without tank or reservoir storage, the capacity of the pumps 
should be such that during the occurrence of fires at least three 
times the mean daily consumption may be furnished. When 
a pump delivers water to a distributing reservoir, its capacity 
need not be so high as in the case of direct pumping, for the reser¬ 
voir storage can be drawn upon in case of fire. When the reser¬ 
voir is large, the pump capacity need be only sufficient to lift the 
annual consumption during the time when it is in operation. The 
subject of pumping is an extensive one, but it will be briefly 
treated from a hydraulic standpoint in Arts. 192 - 201 . 

Gravity supplies are those obtained by impounding the runoff 
of a watershed at an elevation sufficiently high so that the water 
will flow without pumping to the places where it is to be consumed. 
Pumped supplies are obtained either from a stream which lies too 
low to furnish the water by gravity or from the ground from 
water-bearing strata which may be termed natural underground 
reservoirs. Such areas in a sandy country may yield as high as 
i ooo ooo gallons per day per square mile. The borough of 
Brooklyn of the City of New York obtains its water from the 
sands of Long Island, and a good example of the methods to be 
followed in estimating on such a supply is to be found in a 
report by Burr/ Hering, and Freeman.* 

In estimating on the safe yield of a surface watershed a study 
of the existing rainfall and stream flow data should be made. 
In the absence of the latter, estimates of the flow may be made 
by considering the rainfall records and computing the evapora¬ 
tion after allowing for all of the causes by which it is influenced. 
In some cases it will be found that even few rainfall data are 
available, and it then becomes necessary to consider the records 
at the nearest points where such observations have been made, 
and deduce values for the rainfall in the locality being considered.! 
In making estimates of this character all evidence should be 
carefully considered in order to avoid errors. 

* Report on Additional Water Supply, New York, 1903. 
f Monthly Weather Review, March, 1907. 


378 Chap. 11 . Water Supply and Water Power 

When gagings of the stream being studied are available,* 
the problem is a simpler one, but the period during which the 
gagings were taken must be examined with reference to its re¬ 
lation with the rainfall cycle (Art. 142 ). The results shown by 
such a series of gagings during a period of high rainfall would 
differ materially from those during a low cycle. This considera¬ 
tion is of particular importance when determining on the storage 
required for a water supply or for a power plant on a stream of 
moderate size, while on larger streams the controlling factor is 
often simply the quantity and duration of the minimum flow. 
This minimum is generally less dependent on the rainfall cycle 
than is the total yearly yield of the stream. 

Having determined on the quantity of water to be supplied 
and on the flow for a series of years of the stream from which 
the water is to be obtained, it becomes necessary to fix on the 
volume of storage which will be necessary to tide over the driest 
period which is likely to occur. For this purpose the method pro¬ 
posed by Rippl f is a convenient one. It consists essentially in 
determining the net available stream flow for each month, after 
making allowances for evaporation from the reservoir surfaces 
wdiich will result from the new construction and for all other 
possible losses. The total flow for each month is then added to 
the total of the months preceding and since the beginning of the 
period being studied. The total flow from the beginning of the 
period to the end of each month is thus determined and may be 
plotted as in Fig. 145 a. The inclination of the curve AM joining 
the points so plotted thus represents the rate of net available 
stream flow, and may on occasion have a negative value as at 
El, when the evaporation, leakage, and other losses are larger 
than the quantity of water available in the stream. 

The amount of w r ater to be used is now plotted as the line A B, 
it being assumed that the use is at a practically constant rate. 
Wherever the inclination of the curve is greater than that of the 
line A B, the net stream flow is greater than the draft, and wherever 

* Transactions American Society Civil Engineers, vol. 59. 

f Proceedings Institution Civil Engineers, vol. 71. 


Estimates for Water Supply. Art. 145 379 

"it is less the draft is in excess of the available water. To deter¬ 
mine the amount of storage necessary to tide over such a period 
of deficiency, El , if the line EF be drawn parallel to AB and tan¬ 
gent to the curve at E, the maximum ordinate HI will, on the scale 



Fig. 145 a. 

of the diagram, indicate the amount of water which would have 
been necessary to maintain the uniform rate of draft as indicated 
by the line AB. Similarly if AD were the uniform rate of draft, 
the maximum ordinate JK between EG , drawn parallel to AD y 
and the curve would represent the storage volume necessary to 
maintain the draft AD from A to G. The maximum uniform 
rate of draft which could be obtained from A to G would be 
represented by the inclination of the line AC, but this rate, as also 
AB and AD, could not be constantly maintained unless the neces¬ 
sary storage was available at the beginning of the period at A. 
In case the tangent to any summit of the curve and parallel to 
the assumed rate of draft should fail to intersect the curve, it 
would be indicated that the draft was in excess of the total yield 
for the period under consideration. 

Another graphical method is to plot the summation of the monthly 
differences between the net stream flow and the assumed uniform draft. 
In Fig. 1456 if the reservoir be assumed to be full at the beginning 
of the period, then for the next three months the stream flow exceeds 
the draft and an overflow occurs as indicated above the zero line. 





















380 


Chap. 11 . Water Supply and Water Power 


Above this line the actual amount of overflow in each month i^ 
plotted. At the end'of the three months the draft begins to exceed 
the net stream flow and the reservoir level falls, as indicated by the 
continuous line. By the early part of the year 1891 the reservoir has 



again filled. The process is thus continued, and it is found that to 
tide over the period 1890 to 1894, if the reservoir be full at the be¬ 
ginning, a storage capacity of 3 billions of gallons is required. 

The necessary volume of storage having thus been determined, 
it is usual in proportioning the reservoir to make an allowance to cover 
the uncertainties in the data as well as to provide a factor of safety 
against the occurrence of drier years than those covered by the 
records. Such an allowance may range from 10 to 50 percent of 
the storage as determined by the methods of Figs. 145 a and 145 ft. 

The quantity of storage necessary is dependent on the proposed 
rate of draft, but in general it may be said in the northeastern part of 
the United States, on rainfalls of from 38 to 50 inches, that a storage 
capacity of 250 000 000 gallons per square mile of watershed will per¬ 
mit of a safe uniform draft of from 600 000 to 900 000 gallons per square 
mile per day, the smaller figure being applicable to flat watersheds of 
low rainfall and the larger to those which are steep in slope and have 
higher rainfall. 

After the height of the water level of the reservoir is fixed, the 
dimensions of its waste weir may be computed from Arts. 69 and 144 
and the size of the main pipe line by Art. 97 . For the latter com¬ 
putation proper pressures must be assumed throughout the town, so 
that ample head may be provided for fire contingencies. When the 
main divides into branches, the problem of computing the diameters 











































































Estimates for Water Power. Art. 146 


381 


from the given data is indeterminate (Art. 105 ), and hence it will prob¬ 
ably be as well to assume at the outset the sizes of the main and its 
branches. The velocities corresponding to the given quantities and 
the assumed sizes being first computed, the pressure-heads at a num¬ 
ber of points are found. If these are not satisfactory, other sizes are 
to be taken and the computation is to be repeated. The successful 
design will be that which will furnish the required quantities under 
proper pressures with the least expenditure. 

Prob. 145 . How many cubic feet per second per square mile are equiv¬ 
alent to a rainfall of one inch per month ? 

Art. 146 . Estimates for Water Power 

The methods of estimating the water power that can be 
derived by damming a stream are to some extent similar to those 
for water supply. In the absence of gagings the records of rain¬ 
fall and evaporation are to be collected and discussed, but a few 
gagings will probably give more definite information if records of 
water stages during several years can be had. A method of de¬ 
termining the advisable extent of a water power development 
when records of stream flow are available has been developed 
by Herschel.* 

In nearly every situation the stream flow in connection with 
the storage which can be obtained at a reasonable expense is 
not sufficient to continuously generate the power which is re¬ 
quired. In such cases it is necessary to supplement the water 
power with an auxiliary steam plant located at some point within 
the territory to be served where fuel can be obtained most 
economically. In order to determine on the capacity of such an 
auxiliary plant the general method shown in Fig. 145 a may be 
used. With the known volume of available storage and net 
flow of the stream the maximum uniform rate of draft can be 
determined. The capacity of the auxiliary steam plant may 
then be considered as the difference between the power capacity 
required and that furnished by the minimum flow of the stream ; 
while the advisable extent of the water power development will 
depend upon considerations of the river discharge, the cost of 

* Transactions American Society of Civil Engineers, 1907, vol. 58, p. 29. 


382 Chap. 11 . Water Supply and Water Power 

the development, and the cost of installation and operation of 
the auxiliary steam plant. No definite rules are to be laid 
down in this regard, as the exact proportion to be finally decided 
upon depends on many factors which vary in every locality. 

The power needed to be generated by a plant varies from 
hour to hour. The greatest demand is called the “peak.” A 
peak load is one of very short duration and can be met by 
installing an excess of turbine and generator capacity and by 
providing storage in a pond of adequate size. It is probable, 
however, that in many cases the auxiliary heat engines already 
installed to meet low water conditions will more economically 
supply the power for the peak loads than would the necessary 
excess turbine, generator, power house and storage capacity. 

At times of high water the head on the wheels is often re¬ 
duced, due to the change in slope of the river, and the normal 
output of the plant is thus diminished. The “ fall increaser ” 
(Art. 181 ) will operate to increase the available head, or where 
this is not provided the auxiliary steam plant must be called on 
to supply the deficiency. 

Let W be the weight of water delivered per second to a hy¬ 
draulic motor, and h be its effective head as it enters the motor, 
h being due either to pressure (Art. 11 ), or to velocity (Art. 22 ), 
or to pressure and velocity combined (Art. 24 ). The theoretic 
energy per second of this water is 

K=Wh ( 146 ) x 

and if W be in pounds and h in feet, the theoretic horse-power of 
the water as it enters the motor is 

HP =Wh /550 ( 146 ), 

and this is the power that can be developed by a motor of effi¬ 
ciency unity. The work k delivered by the motor is, however, 
always less than K, owing to losses in impact and friction, and the 
horse-power hp of the motor is less than HP. The efficiency 
of the motor is 


e = k/K = k/Wh or e = hp/HP 


(146)3 




Estimates for Water Power. Art. 146 


383 


and the value of this for turbine wheels is usually about 0.80; 
that is, the wheel transforms into useful work about 80 percent 
of the energy of the water that enters it. 

In designing a water-power plant it should be the aim to ar¬ 
range the forebays and penstocks which lead the water to the 
wheel so that the losses in these approaches may be as small as 
possible. The entrance from the head race into the forebay, 
from the forebay into the penstock, and from the penstock to the 
motor should be smooth and well rounded; sudden changes in 
cross-section should be avoided, and all velocities should be low 
except that at the motor. If these precautions be carefully ob¬ 
served, the loss of head outside of the motor can be made very 
small. Let H be the total head from the water level in the head 
race to that in the tail race below the motor. The total available 
energy per second is WH , and it should be the aim of the designer 
to render the losses of head in the approaches as small as possible 
so that the effective head h may be as nearly equal to H as pos¬ 
sible. Neglect of these precautions may render the effective 
power less than that estimated. 

The efficiency e\ of the approaches is the ratio of the energy 
K of the water as it enters the wheel to the maximum available 
energy WH, or e\ = K/WH. The efficiency e of the entire plant, 
consisting of both approaches and wheel, is the ratio of the work k 
delivered by the wheel to the energy WH, or 

e = k/WH = eK/WH = ee\ 

or, the final efficiency is the product of the separate efficiencies. 
If the efficiency of the wheel be 0.75 and that of the approaches 
0.96, the efficiency of the plant as a whole is 0.72, or only 72 per¬ 
cent of the theoretic energy is utilized. Usually the efficiency 
of the approaches can be made higher than 96 percent. 

In making estimates for a proposed plant, the efficiency of 
turbine wheels may generally be taken at 80 percent; the effec¬ 
tive work is then 0.80IT//, and accordingly if the wheels are 
required to deliver the work k per second, the approaches are to 
be so arranged that Wh shall not be less than 1.25&. Especially 


384 Chap. 11. Water Supply and Water Power 

when the water supply is limited it is important to make all 
efficiencies as high as possible. 

Prob. 146 . A stream delivers 500 cubic feet of water per second to a 
canal which terminates in a forebay where the water level is 8.1 feet above 
the tail race. The wheels deliver 335 horse-power and their efficiency is 
known to be 75 percent. How much power is lost in the forebay and pen¬ 
stock ? 

Art. 147. Water delivered to a Motor 

To determine the efficiency of a hydraulic motor by formula 
(146) 3 the effective work k is to be measured by the methods of 
Art. 149^ and the head h to be ascertained by Art. 148. In order 
to find the weight W that passes through the wheel in one second, 
there must be known the discharge per second q and the weight 
w of a cubic unit of water; then 

W = wq 

Here w may be found by weighing one cubic foot of the water, 
or when the water contains few impurities its temperature may 
be noted and the weight be taken from Table 3. In approximate 
computations w may be taken at 62.5 pounds per cubic foot. In 
precise tests of motors, however, its actual value should be ascer¬ 
tained as closely as possible. 

The measurement of the flow of water through orifices, weirs, 
tubes, pipes, and channels has been so fully discussed in the pre¬ 
ceding chapters, that it only remains here to mention one or two 
simple methods applicable to small quantities, and to make a 
few remarks regarding the subject of leakage. In any particular 
case that method of determining q is to be selected which will 
furnish the required degree of precision with the least expense. 

For a small discharge the water may be allowed to fall into a 
tank of known capacity. The tank should be of uniform horizontal 
cross-section, whose area can be accurately determined, and then 
the heights alone need be observed in order to find the volume. 
These in precise work will be read by hook gages, and in cases of 
less accuracy by measurements with a graduated rod. At the 
beginning of the experiment a sufficient quantity of water must be 
in the tank so that a reading of the gage can be taken; the water 


Water delivered to a Motor. Art. 147 


385 


is then allowed to flow in, the time between the beginning and end 
of the experiment being determined by a stop-watch, duly tested and 
rated. This time must not be short, in order that the slight errors 
in reading the watch may not affect the result. The gage is read 
at the close of the test after the surface of the water becomes quiet, 
and the difference of the gage readings gives the depth which has 
flowed in during the observed time. The depth multiplied by the area 
of the cross-section of the tank gives the volume, and this divided 
by the number of seconds during which the flow has occurred fur¬ 
nishes the discharge per second q. 

If the discharge be very small, it may be advisable to weigh the 
water rather than to measure the depths and cross-sections. The total 
weight divided by the time of flow then gives directly the weight W. 
This has the advantage of requiring no temperature observation, 
and is probably the most accurate of all methods, but unfortunately 
it is not possible to weigh a considerable volume of water except at 
great expense. 

When water is furnished to a motor through a small pipe, a com¬ 
mon water meter may often be advantageously used to determine 
the discharge (Art. 38). No water meter, however, can be regarded 
as accurate until it has been tested by comparing the discharge as re¬ 
corded by it with the actual discharge as determined by measurement 
or weighing in a tank. Such a test furnishes the constants for cor¬ 
recting the result found by its readings, which otherwise is liable 
to be 5 or io percent in error. The Venturi meter (Art. 38) fur¬ 
nishes an accurate method of measuring large quantities. 

The leakage which occurs in the flume or penstock before the water 
reaches the wheel should not be included in the value of W, which is 
used in computing its efficiency, although it is needed in order to as¬ 
certain the efficiency of the entire plant. The manner of determining 
the amount of leakage will vary with the particular circumstances of 
the case in hand. If it be small, it may be caught in pails and directly 
weighed. If large in quantity, the gates which admit water to the 
w r heel may be closed, and the leakage being then led into the tail race, 
it may be there measured by a weir, or by allowing it to collect in a 
tank. The leakage from a vertical penstock whose cross-section is 
known may be ascertained by filling it with water, the wheel being 
still, and then observing the fall of the water level at regular intervals 
of time. In designing constructions to bring water to a motor, it is 


386 


Chap. 11. Water Supply and Water Power 


best, of course, to arrange them so that all leakage will be avoided, but 
this cannot always be fully attained, except at great expense. 

The most common method of measuring q is by means of a weir 
placed in the tail race below the wheel. This has the disadvantage 
that it sometimes lessens the fall which would be otherwise available, 
and that often the velocity of approach is high. It has, however, the 
advantage of cheapness in construction and operation, and for any 
considerable discharge appears to be almost the only method which is 
both economical and precise. If the weir is placed above the wheel, 
the leakage of the penstock must be carefully ascertained. 

Prob. 147 . A weir w'ith end contractions and no velocity of approach 
has a length of 1.33 feet, and the depth on the crest is 0.406 feet. The same 
water passes through a small turbine under the effective head 10.49 feet. 
Compute the theoretic horse-power. 

Art. 148 . Effective Head on a Motor 

The total available head H between the surface of the water 
in the reservoir or head race and that in the lower pool or tail 
race is determined by running a line of levels from one to the other. 
Permanent bench marks being established, gages can then be 
set in the head and tail races and graduated so that their zero 
points will be at some datum below the tail-race level. During 
the test of a wheel each gage is read by an observer at stated 
intervals, and the difference of the readings gives the head II. 
In some cases it is possible to have a floating gage on the lower 
level, the graduated rod of which is placed alongside a glass tube 
that communicates with the upper level; the head II is then 
directly read by noting the point of the graduation which coin¬ 
cides with the water surface in the tube. This device requires 
but one observer, while the former requires two; but it is usually 
not the cheapest arrangement unless a large number of observa¬ 
tions are to be taken. 

From this total head H are to be subtracted the losses of head 
in entering the forebay and penstock, and the loss of head in 
friction in the penstock itself, and these losses may be ascertained 
by the methods of Chaps. 8 and 9. Then 

h = H — ti -h" 


Effective Head on a Motor. Art. 148 


387 


is the effective head acting upon the wheel. In properly designed 
approaches the lost heads h' and h" are very small. 

When water enters upon a wheel through an orifice which is 
controlled by a gate, losses of head will result, which can be 
estimated by the rules of Chaps. 5 and 6. If this orifice is in 
the head race, the loss of head should be subtracted together 
with the other losses from the total head H. But if the regulating 
gates are a part of the wheel itself, as is the case in a turbine, the 
loss of head should not be subtracted, because it is properly 
chargeable to the construction of the wheel, and not to the ar¬ 
rangements which furnish the supply of water. In any event that 
head should be determined which is to be used in the subsequent 
discussions : if the efficiency of the fall is desired, the total avail¬ 
able head is required ; if the efficiency of the motor, that effective 
head is to be found which acts directly upon it (Art. 146). 

When water is delivered through a nozzle or pipe to an im¬ 
pulse wheel, the head h is not the total fall, since a large part of 
this may be lost in friction in the pipe, but is merely the velocity- 
head v 2 / 2 g of the issuing jet. The value of v is known when the 
discharge q and the area of the cross-section of the stream have 
been determined, and 

It = v 2 / 2 g = (q/a) 2 / 2 g 

In the same manner when a stream flows in a channel against 
the vanes of an undershot wheel the effective head is the velocity- 
head, and the theoretic energy is 

K = Wh = Wv 2 / 2 g = wq 3 f 2 ga 2 

If, however, the water in passing through the wheel falls a dis¬ 
tance ho below the mouth of the nozzle, then the effective head 
which acts upon the wheel is given by 

h = v 2 / 2 g + ho 

In order to fully utilize the fall ho it is plain that the wheel should 
be placed as near the level of the tail race as possible. 

Lastly, when water enters a turbine wheel through a pipe, 
a piezometer may be placed near the wheel entrance to register 
the pressure-head during the flow; if this pressure-head, meas- 


388 Chap. 11. Water Supply and Water Power 

ured+upon and from the water level in the tail race, be called h\ 
and if the velocity in the pipe be v, then 

h = h\ + v 2 / 2 g 

is the effective head acting on the wheel. It is here supposed 
that the turbine has a draft tube leading below the water level 
in the tail race; if this is not the case, li\ should be measured 
upward from the lowest part of the exit orifices. 

Prob. 148 . A pressure gage at the entrance of a nozzle registers 116 
pounds per square inch, and the coefficient of velocity of the nozzle is 0.98. 
Compute the effective velocity-head of the issuing jet. 

Art. 149 . Measurement of Effective Power 

The effective work and horse-power delivered by a water¬ 
wheel or hydraulic motor is often required to be measured. 
Water power may be sold by means of the weight W, or quantity 
q, furnished under a certain head, leaving the consumer to pro¬ 
vide his own motor; or it may be sold directly by the number 
of horse-power. In either case tests must be made from time to 
time in order to insure that the quantity contracted for is actually 
delivered and is not exceeded. It is also frequently required to 
measure effective work in order to ascertain the power and effi¬ 
ciency of the motor, either because the party who buys it has 
bargained for a certain power and efficiency, or because it is 
desirable to know exactly what the motor is doing in order to 
improve if possible its performance. 

The test of a hydraulic motor has for its object: first, the 
determination of the effective energy and power; second, the 
determination of its efficiency; and third, the determination of 
that speed which gives the greatest power and efficiency. If the 
wheel be still, there is no power; if it be revolving very fast, the 
water is flowing through it so as to change but little of its energy 
into work: and in all cases there is found a certain speed which 
gives the maximum power and efficiency. To execute these tests, 
it is not at all necessary to know how the motor is constructed 
or the principle of its action, although such knowledge is very 


Measurement of Effective Power. Art. 149 


389 


valuable, and is in fact indispensable to enable the engineer to 
suggest methods by which its operation may be improved. 

A method in which the effective work of a small motor may 
be measured is to compel it to exert all its power in lifting a weight. 
For this purpose the weight may be attached to a cord which 
is fastened to the horizontal axis of the motor, and around which 
it winds as the shaft revolves. The wheel then expends all its 
power in lifting this weight W\ through the height hi in ti seconds, 
and the work performed per second then is k = W\hi/t\. This 
method is rarely used in practice on account of the difficulty of 
measuring h with precision. 

The usual method of measuring the effective work of a hy¬ 
draulic motor is by means of the friction brake or power dyna¬ 
mometer invented by Prony 
about 1780 . In Fig. 149 is illus¬ 
trated a simple method of apply¬ 
ing the apparatus to a vertical 
shaft, the upper diagram being 
a plan and the lower an eleva¬ 
tion. Upon the vertical shaft 
is a fixed pulley, and against 
this are seen two rectangular 
pieces of wood hollowed so as 
to fit it, and connected by two 
bolts. By turning the nuts on 
these bolts while the pulley is 
revolving, the friction can be in¬ 
creased at pleasure, even so as 
to stop the motion; around these bolts between the blocks are 
two spiral springs (not shown in the diagram) which press the 
blocks outward when the nuts are loosened. To one of these 
blocks is attached a cord which runs horizontally to a small 
movable pulley over which it passes, and supports a scale-pan 
in which weights are placed. This cord runs in a direction op¬ 
posite to the motion of the shaft, so that when the brake is 
tightened, it is prevented from revolving by the tension caused 






































390 Chap. 11. Water Supply and Water Power 

by the weights. The direction of the cord in the horizontal 
plane must be such that the perpendicular let fall upon it from 
the center of the shaft, or its lever-arm, is constant; this can be 
effected by keeping the small pointer on the brake at a fixed 
mark established for that purpose. 

To measure the work done by the wheel, the shaft is discon¬ 
nected from the machinery which it usually runs, and allowed to 
revolve, transforming all its work into heat by the friction be¬ 
tween the revolving pulley and the brake, which is kept stationary 
by tightening the nuts, and at the same time placing sufficient 
weights in the scale-pan to hold the pointer at the fixed mark. 
Let n be the number of revolutions per second as determined 
by a counter attached to the shaft, P the tension in the cord which 
is equal to the weight of the scale-pan and its loads, l the lever- 
arm of this tension with respect to the center of the shaft, r the 
radius of the pulley, and F the total force of friction between the 
pulley and the brake. Now in one revolution the force F is over¬ 
come through the distance 277y, and in n revolutions through the 
distance 2 rrrn. Hence the effective work done by the wheel 
in one second is 

k = F ° 2 'rrrn = 27 rn • Fr 

The force F acting with the lever-arm r is exactly balanced by the 
force P acting with the lever-arm l ; accordingly the moments 
Fr and PI are equal, and hence the work done by the wheel in 
one second is 

If P is in pounds and l in feet, the effective horse-power of the 

wheel is given by 77 , 

J hp = 2 irnPl/$$o 

As the number of revolutions in one second cannot be accurately 
read, it is usual to record the counter readings every minute or 
half-minute; if A be the number of revolutions per minute, 

hp = 2 'ttNPI /33 000 (149)2 

It is seen that this method is independent of the radius of the 
pulley, which may be of any convenient size ; for a small motor the 
brake may be clamped directly upon the shaft, but for a large 


k = 27 mPl 


(149)1 


Measurement of Effective Power. Art. 149 391 

one a pulley of considerable size is needed, and a special arrange¬ 
ment of levers is used instead of a cord. 

The efficiency of the motor is now found by dividing the effec¬ 
tive work per second by the theoretic work per second. Let K 
be this theoretic work, which is expressed by Wh, where W and 
h are determined by the methods of Arts. 147 and 148; then 

e = k/K or 6 = hp/HP 

The work measured by the friction brake is that delivered at the 
circumference of the pulley, and does not include that power 
which is required to overcome the friction of the shaft upon its 
bearings. The shaft or axis of every water-wheel must have at 
least two bearings, the friction of which consumes probably about 
2 or 3 percent of the power. The hydraulic power and efficiency 
of the wheel, regarded as a user of water, are hence 2 or 3 percent 
greater than the values computed from above formulas. For 
example, let P = 12.5 pounds, /= 14.31 feet, and ^ = 635 , then 
21.6 horse-powers are in total delivered by the wheel, of which 
about 0.6 horse-power is consumed in shaft friction. 

There are in use various forms and varieties of the friction brake, 
but they all act upon the principle and in the manner above described. 
For large wheels they are made of iron, and almost completely encircle 
the pulley; while a special arrangement of levers is used to lift the large 
weight P* If the work transformed into friction be large, both 
the brake and the pulley may become hot, to prevent which a stream 
of cool water is allowed to flow upon them. To insure steadiness of 
motion, it is well that the surface of the pulley should be lubricated, 
which for a wooden brake is well done by the use of soap. It is impor¬ 
tant that the connection of the cord to the brake should be so made 
that the lever-arm I increases when the brake moves slightly with the 
wheel; if this is not done, the equilibrium will be unstable and the 
wheel will be apt to cause the brake to revolve with it. 

Prob. 149 . Find the power and efficiency of a motor when the theoretic 
energy is 1.38 horse-power, which makes 670 revolutions per minute, the 
weight on the brake being 2 pounds 14 ounces and its lever-arm 1.33 feet. 

* Thurston, in Transactions American Society of Mechanical Engineers, 
1886, vol. 8, p. 359. 




392 


Chap. 11. Water Supply and Water Power 


Art. 150. Tests of Turbine Wheels 

The following description of a test of a 6 -inch Eureka turbine, 
made in 1888 at the hydraulic laboratory of Lehigh University, 
may serve to exemplify the methods of the preceding articles. 
The water was measured by a weir from which it ran into a verti¬ 
cal penstock 15.98 square feet in horizontal cross-section. This 
plan of having the weir above the wheel is not a good one, but it 
was here adopted on account of lack of room below the turbine. 
When a constant head was maintained in the penstock, the quan¬ 
tity of water flowing through the wheel was the same as. that pass¬ 
ing the weir; if, however, the head in the penstock fell x feet per 
minute, the flow through the wheel in cubic feet per minute was 
60 q + 15 . 98 :*;, in which q is the discharge per second over the 
weir. As the supply of water was very limited, the wheel could 
not be run to its fully capacity. The level of water in the pen¬ 
stock was read upon a head gage consisting of a glass tube behind 
which a graduated scale was fixed, the zero of which was a little 
above the water level in the tail race. The latter level was read 
upon a fixed graduated scale having its zero in the same horizon¬ 
tal plane as the first; these readings were hence essentially nega¬ 
tive. The head upon the wheel is then found by adding the read¬ 
ings of the two gages. 

The vertical shaft of the turbine, being about 15 feet long, 
was supported by four bearings, and to a small pulley upon its 


Time on 
April 13, 
1888 

Depth 
on Weir 
Crest 
Feet 

Penstock 

Gage 

Feet 

Tail-race 

Gage 

Feet 

j 7 m 

0.288 

II.25 

— 0.21 

18 

0.289 

II.17 

0.20 

19 

0.289 

II .13 

0.21 

20 

0.288 

II.IO 

0.21 

3I1 2 2 m 

0.287 

IO.81 

— 0.20 

23 

0.287 

IO.69 

0.20 

24 

0.287 

IO.62 

0.21 

25 

0.286 

IO.57 

0.21 


Revolu¬ 
tions in 
One 
Minute 

Weight 

on 

Brake 

Pounds 

Remarks 

635 

2-5 

Length of weir, 

625 

2-5 

b = 1.909 feet. 

635 

2-5 

Length of lever- 


2.5 

arm on brake, 



l = 1.431 feet. 

535 

3-o 

Gate of wheel | 

540 

3-o 

open during all ex- 

535 

3-° 

periments. 


3-o 

















Tests of Turbine Wheels. Art. 150 


393 


upper end was attached the friction dynamometer, as described 
in the last article. The number of revolutions was read from a 
counter placed in the top of this shaft. The observations were 
taken at minute intervals, electric bells giving the signals, so 
that precisely at the beginning of each minute simultaneous read¬ 
ings were taken by observers at the weir, at the head gage, at 
the tail gage, and at the counter, the operator at the brake con¬ 
tinually keeping it in equilibrium with the resisting weight in the 
scale-pan by slightly tightening and loosening the nuts as required. 
The above shows notes of all the observations of two sets of tests, 
each lasting three minutes, the weight in the scale-pan being 
different in the two sets. 

The following are the results of the computations made from 
the above notes for each minute interval. The second column 
is derived from formula (63)i, using the coefficient corresponding 
to the given length of weir and depth on crest. The third column 
is obtained by taking the differences of the observed readings of 
the penstock head gage. The fourth column gives the discharge 


Interval 

of 

Time 

Discharge 

over 

Weir 

{Cubic Feet 
per 

Minute 

Fall in 
Penstock 
Feet 

Flow 
through 
Wheel 
Cubic Feet 
per 

Minute 

Head on 
Wheel 
Feet 

Theoretic 
Horse¬ 
power of 
the 
Water 

Effective 
Horse¬ 
power of 
the 
Wheel 

Efficiency 
of the 
Wheel 
Percent 

I7 ra to i8 m 

58.49 

+ 0.08 

59-77 

II.41 

I.290 

0-433 

33-6 

18 to 19 

58.66 

+ 0.04 

59-30 

II.36 

I.274 

0.426 

33-4 

19 to 20 

58.49 

+ O.03 

58.97 

11.32 

1.262 

0-433 

34-3 

22 m to 23 m 

58.05 

+ 0.13 

60.13 

IO.95 

1-245 

0-437 

35 -i 

23 to 24 

58.05 

+ 0.07 

59-17 

10.86 

1.215 

O.441 

36.3 

24 to 25 

57-88 

+ 0.05 

58.68 

10.80 

1.198 

0-437 

36.5 


Q through the wheel found as above explained. The fifth column 
is the mean head h on the wheel during the minute, as derived from 
the observed readings of head and tail gage. The sixth column 
is found by formula (146) 2 , using for W its value q\wQ, in which 
w is taken at 62.4 pounds per cubic foot. The seventh column 
is computed from formula (149) 2 ; and the last column is found 
























394 Chap. 11. Water Supply and Water Power 

by dividing the numbers in the seventh by those in the sixth 
column. 

These results show that the mean efficiency of the wheel and 
shaft under the conditions stated was about 35 percent; this low 
figure being due to the circumstance that the gate was not fully 
opened. It is also seen that the mean efficiency of the second set 
is 2.2 percent greater than that of the first set; this is due to 
the lower speed, and with still lower speeds the efficiency was found 
to be lower, so that a speed of about 535 revolutions per minute 
gives the maximum efficiency. 

The work of Francis on the experiments made by him at 
Lowell, Mass., will always be a classic in American hydraulic 
literature, for the methods therein developed for measuring the 
theoretic power of a waterfall and the effective power utilized 
by the wheel are models of careful and precise experimentation.* 
In determining the speed of the wheel he used a method somewhat 
different from that above explained, namely, the counter attached 
to the shaft was connected with a bell which struck at the com¬ 
pletion of every 50 revolutions; the observer at the counter had 
then only to keep his eye upon the watch, and to note the time at 
certain designated intervals, say at every sixth stroke of the 
bell. The number of revolutions per second was then obtained 
by dividing the number of revolutions in the interval by the num¬ 
ber of seconds, as determined by the watch. This method re¬ 
quires a stop-watch in order to do good work, unless the observer 
be very experienced, as an error of one second in an interval of 
one minute amounts to 1.7 percent. 

At Holyoke, Mass., there is a permanent flume for testing 
turbines arranged with a weir which can be varied up to lengths of 
20 feet, so as to test the largest wheels which are constructed. 
As the expense of fitting up the apparatus for testing a large tur¬ 
bine at the place where it is to be used is often great, it is some¬ 
times required in contracts that the wheel shall be sent to a place 
where a special outfit for such work exists. The wheel is mounted 
in the testing flume, and there, by the methods explained in the 

* Lowell Hydraulic Experiments, 1 st Edition, 1855 ; 4 th, 1883 . 


Tests of Turbine Wheels. Art. 150 


395 


preceding articles, it is run at different speeds in order to deter¬ 
mine the speed which gives the maximum efficiency as well as the 
effective power developed at each speed. As the efficiency of a 
turbine varies greatly with the position of the gate which admits 
the water to it, tests are made with the gate fully opened and at 
various partial openings. The results thus obtained are not only 
valuable in furnishing full information concerning the effective 
power and efficiency of the wheel, but they also convert the turbine 
into a water meter, so that when running under the same head as 
during the tests, the quantity of water which passes through it 
per second can at any time be closely ascertained by noting the 
number of revolutions per second. 

The following gives the results of the tests of an 8 o-inch outward- 
flow Boyden turbine, made at Holyoke in 1885 , the gate being fully 
opened in each experiment. The heads in the second column were 
derived from the head and tail race gages, these being arranged so 


Number 

Head 
in Feet 

Revolutions 

per 

Minute 

Discharge 
Cubic Feet 
per Second 

Horse-power 

Efficiency 

Percent 

21 

17.16 

63-5 

I17.OI 

172.57 

1-0 

00 

VO 

20 

17.27 

70.0 

118.37 

177.41 

76.60 

19 

17-33 

75-o 

H9-53 

178.63 

76.11 

l8 

17-34 

80.0 

I2I.I5 

178.32 

74.92 

17 

17.21 

86.0 

122.41 

178.57 

74.81 

l6 

17.21 

93-2 

124.74 

176.44 

72.54 

15 

17.19 

100.0 

127-73 

167.94 

67-51 


that one observer could directly read the difference. The numbers 
in the third column were found by dividing the total number of revo¬ 
lutions during the experiment by its length in minutes; those in the 
fourth by the weir formula (63)!; those in the fifth by (149) 2 from 
the records of the friction dynamometer; and those in the last column 
were computed by (146) 3 . It is seen that the discharge always in¬ 
creased with the speed of the wheel, and the reason for this is explained 
in Art. 166. The maximum efficiency of 76.6 percent occurred at 70 
revolutions per minute; and for 100 revolutions per minute the effi¬ 
ciency was lowered to 67.7 percent, notwithstanding that the quantity 
of water passing through the wheel was much greater. 














396 


Chap. 11 . Water Supply and Water Power 


Prob. 150 . Compute the theoretic horse-power and the efficiency for 
the above experiments, Nos. 15 and 21, on the large Boyden outward-flow 
turbine. 


Art. 151. Facts concerning Water Power 

The number of horse-powers generated by water-wheels and 
turbines and used in manufacturing establishments in the United 
States was 1 130 431 in 1870, 1 225 379 in 1880, 1 263 343 in 1890, 
and 1 727 258 in 1900; these figures do not include the electric 
power derived from water. In 1908* the total development 
was 5 356 680 horse-powers in 52 827 wheels and turbines. 
Since 1890 there has been a large development of water power 
in connection with electric light and trolley service, and this 
development promises to attain great proportions during the 
twentieth century. It has been estimated that the rivers of the 
United States can furnish about 212 000 000 horse-powers, so that 
the possibilities for the future are almost unlimited. 

Water power is sometimes sold by what is called the “mill 
power,” which may be roughly supposed to be such a quantity 
as the average mill requires, but which in any particular case must 
be defined by a certain quantity of water under a given head. 
Thus at Lowell the mill power is 30 cubic feet per second under 
a head of 25 feet, which is equivalent to 85.2 theoretic horse¬ 
power. At Minneapolis it is 30 cubic feet per second, under 22 
feet head, or 75 theoretic horse-power. At Holyoke it is 38 cubic 
feet per second under 20 feet head, or 86.4 theoretic horse-power. 
This seems an excellent way to measure power when it is to be 
sold or rented, as the head in any particular instance is not subject 
to much variation; or if so liable, arrangements must be adopted 
for keeping it nearly constant, in order that the machinery in the 
mill may be run at a tolerably uniform rate of speed. Thus 
nothing remains for the water company to measure except the 
water used by the consumer. The latter furnishes his own motor, 
and is hence interested in securing one of high efficiency, that he 
may derive the greatest power from the water for which he pays. 
The perfection of American turbines is undoubtedly largely due 

* Water Supply and Irrigation Paper, No. 234. 


Facts concerning Water Power. Art. 151 397 

to this method of selling power, and the consequent desire of the 
mill owners to limit their expenditure for water. The turbine 
itself, when tested and rated, becomes a meter by which the com¬ 
pany can at any time determine the quantity of water that passes 
through it. 

A common method of selling the power which is generated 
by turbines is by the nominal horse-power of the wheel as stated 
in the catalogue of the manufacturer. The seller fixes a price per 
annum for one horse-power on this basis, and the buyer furnishes 
his own wheel. By this method no controversy can arise regard¬ 
ing the amount of water used, for the purchaser has the right to 
use all that can pass through the turbine. The head to be used 
for finding the nominal horse-power is the mean head which can 
be utilized by the wheel, and this must be agreed upon in advance 
between the parties. 

The power of electric generators is usually expressed in kilo¬ 
watts. One English horse-power in 0.746 kilowatts, and one 
metric horse-power is 0.736 kilowatts. One kilowatt is 1.340 
English horse-powers or 1.360 metric horse-powers. The effi¬ 
ciency of a good electric generator is about 95 percent, so that it 
delivers 95 percent of the work imparted to it by the turbine 
wheel; if the efficiency of this wheel is 75 percent, the combined 
efficiency of the hydraulic and electric plant is 71 percent. Elec¬ 
tric power is usually sold by the kilowatt-hour, this being meas¬ 
ured by a wattmeter. 

The available power of natural waterfalls is very great, but it is 
probably exceeded by that which can be derived from the tides and 
waves of the ocean. Twice every day, under the attraction of the 
sun and moon, an immense weight of water is lifted, and it is theoret¬ 
ically possible to derive from this a power due to its weight and lift. 
Continually along every ocean beach the waves dash in roar and foam, 
and energy is wasted in heat which by some device might be utilized 
in power. The expense of deriving power from these sources is indeed 
greater than that of the water wheel under a natural fall, but the time 
may come when the profit will exceed the expense, and then it will cer¬ 
tainly be done. Coal and wood and oil may become exhausted, but 
as long as the force of gravitation exists, and the ocean remains upon 


398 


Chap. 11 . Water Supply and Water Power 


which it can act, power, heat, and light can be generated in unlimited 
quantities. 

Prob. 151 a. Deduce the simple and useful rule that one inch of rainfall 
per hour is, very nearly, equivalent to one cubic foot per second per acre. 

Prob. 1515 . Find the theoretic horse-power of a plant where 1200 cubic 
feet of water per second is used under a total head of 49.5 feet. If the 
efficiency of the approaches is 99 per cent, the efficiency of the turbines 
76 percent, and the efficiency of the dynamos 96 percent, what power in 
kilowatts is delivered ? 

Prob. 151 c. What is the theoretic metric horse-power of a plant where 
112 cubic meters of water per second are used under a head of 23.5 meters? 
If the efficiencies of the approaches, turbines, and electric generators are 
98.5, 74.3, and 97.5 percent, respectively, compute the number of metric 
horse-powers delivered, and also the power in kilowatts. 

Prob. 151 d. When a turbine is tested by a friction dynamometer, show 
that its power in kilowatts is o.ooio3iVP/, if P be the load on the brake in 
kilograms, / its lever-arm in meters, and N the number of revolutions per 
minute. When N = 200, P = 250 kilograms, and / = 2.01 meters, what 
electric power is delivered by a dynamo attached to the turbine when the 
efficiency of the dynamo is 97.2 percent ? 

Prob. 151 c. The hectare-meter is a convenient unit for estimating 
large quantities of water in irrigation and water-supply work. Show that 
one hectare-meter is 10 000 cubic meters. Show that 100 centimeters of 
rainfall falling in one month is, very nearly, 0.004 cubic meters per second 
per hectare. 



Definitions and Principles. Art. 152 


399 


CHAPTER 12 

DYNAMIC PRESSURE OF WATER 

Art. 152 . Definitions and Principles 

The pressures exerted by moving water against surfaces which 
change its direction or check its velocity are called dynamic, 
and they follow very different laws from those which govern the 
static pressures that have been discussed and used in the preceding 
chapters. A static pressure due to a certain head may cause a 
jet to issue from an orifice; but this jet in impinging upon a 
surface may cause a dynamic pressure less than, equal to, or 
greater than that due to the head. A static pressure at a given 
point in a mass of water is exerted with equal intensity in all direc¬ 
tions ; but a dynamic pressure is exerted in different directions 
with different intensities. In the following chapters the words 
“ static ” and “ dynamic ” will generally be prefixed to the word 
“ pressure,” so that no confusion may result. 

The dynamic pressure exerted by a stream flowing with a 
given velocity against a surface at rest is evidently equal to that 
produced when the surface moves in still water with the same 
velocity. This principle was applied in Art. 40 in rating the 
current meter, the vanes of which move under the impulse of the 
impinging water. The dynamic pressure exerted upon a moving 
body by a flowing stream depends upon the velocity of the body 
relative to the stream. 

The “impulse” of a jet or stream of water is defined as the 
dynamic pressure which it is capable of producing in the direction 
of its motion when its velocity is entirely destroyed in that direc¬ 
tion. This can be done by deflecting the jet normally sidewise by 
a fixed surface; when the surface is smooth, so that no energy 
is lost in frictional resistances, the actual velocity remains un- 


400 


Chap. 12 . Dynamic Pressure of Water 


altered, but the velocity in the original direction has been ren¬ 
dered null. In Art. 27 it is shown that the theoretic force of 
impulse of a stream of cross-section a and velocity v is 

F = W - = wq V = 2wa — ( 152 ) 

g ^g ig 


in which W and q are the weight and volume delivered per second, 
and w is the weight of one cubic unit of water. This equation 
shows that the dynamic pressure that may be produced by im¬ 
pulse is equal to the static pressure due 
to twice the head corresponding to the 
velocity v. It would then be expected, 
when two equal orifices or tubes are 
placed exactly opposite, as in Fig. 125 , 
and a loose plate is placed verti¬ 
cally against one of them, that the 
dynamic pressure upon the plate caused 
by the impulse of the jet issuing from A 
under the head h would balance the static pressure caused by 
the head 2 h. This conclusion has been confirmed by experiment, 
for a tube A which has a smooth inner surface and rounded 
inner edges so that its coefficient of discharge is unity. 



The reaction of a jet or stream is the backward dynamic 
pressure, in the line of its motion, which is exerted against a 
vessel out of which it issues, or against a surface away from which 
it moves. This is equal and opposite to the impulse, and the 
equation above given expresses its value and the laws which 
govern it. The expression for the reaction or impulse F in ( 152 ) 
may be also proved as follows: The definition by which forces 
are compared with each other is, that forces are proportional to 
the accelerations which they can produce. The weight W , if 
allowed to fall, acquires the acceleration g ; the force F which can 
produce the acceleration v is hence related to W and g by the 
equation F/W = v/g, and accordingly F = W • v/g. 


The forces of impulse and reaction do not really exist in a stream 
flowing with constant velocity and direction, although F indicates 
the force that was exerted in putting the stream into motion and the 
































Experiments on Impulse and Reaction. Art. 153 401 

force that is required to stop it. When the direction of the stream is 
changed by opposing obstacles, the impulse and reaction produce 
dynamic pressure; if, in making this change, the absolute velocity is 
retarded, energy is converted into work. Impulse and reaction are of 
practical value, because the resulting dynamic pressures may be uti¬ 
lized for the production of work. For this purpose water is made 
to impinge upon moving vanes, which alter both its direction 
and velocity, thus producing a dynamic pressure P, which overcomes 
in each second an equal resisting force through the space u. The work 
done per second is then k = Pu, and it is the object in designing a hy¬ 
draulic motor to make this work as large as possible; for this purpose, 
the most advantageous values of P and u are to be selected. 

The word “impact” is sometimes popularly used to designate 
impulse or pressure, but in hydraulics it refers to those cases where 
energy is lost in eddies and foam, as when a jet impinges into water 
or upon a rough plane surface. Impact is not defined in algebraic 
terms, but the energy lost in impact may be so defined and computed. 
When the energy of a stream of water is to be utilized, losses due to 
impact should be avoided. Whenever impact occurs, kinetic energy 
is transformed into heat. 

Prob. 152 . When a jet is one inch in diameter, how many gallons 
per second must it deliver in order that its impulse may be ioo pounds ? 


Art. 153 . Experiments on Impulse and Reaction 


A simple device by which the dynamic pressure P exerted 
upon a surface by the impulse and reaction of a jet that glides 
over it can be directly weighed is 
shown in Fig. 153 a. It consists 
merely of a bent lever supported 
on a pivot at O, and having a plate 
A attached at the lower end of the 
vertical arm upon which the stream 
impinges, while weights applied at 
the end of the other arm measure 
the dynamic pressure produced by the impulse. By means of an 
apparatus of this nature, experiments have been made by Bidone, 
Weisbach, and others, the results of which will now be stated. 




Fig. 153 a. 











402 


Chap. 12 . Dynamic Pressure of Water 


When the surface upon which the stream impinges is a plane 
normal to the direction of the stream, as shown at A, the dynamic 
pressure P closely agrees with that given by the theoretic formula 
for F in the last article, namely, 

P = W- = 2 wa — ( 153 ) 

g 2 £ 

being about 2 percent greater according to Bidone, and about 
4 percent less according to Weisbach. The actual value of P 
was found to vary somewhat with the size of the plate, and with 
its distance from the end of the tube from which the jet issued. 

When the surface upon which the stream impinges is curved, 
as at B, or so arranged that the water is turned backward from 
the surface, the value of the dynamic pressure P was found to be 
always greater than the theoretic value, and that it increased 
with the amount of backward inclination. When a complete 
reversal of the original direction of the water was obtained, as 
at C, it was found that P, as measured by the weights, was nearly 
double the value of that against the plane. This is explained by 
stating that as long as the direction of the flow is toward the sur¬ 
face the dynamic pressure of its impulse is exerted upon it, but 
when the water flows backward away from the surface, the 
dynamic pressure due to both impulse and reaction is then 
exerted upon it. The sum of these is 

P=F + F=2W - =4 wa — 

g 2g 

which agrees with the results experimentally obtained. 

An experiment by Morosi * shows clearly that the dynamic 
pressure against a surface may be increased still further by the 
device shown in Fig. 1536 , where the stream is made to perform 
two complete reversals upon the surface. He found that in this 
case the value of the dynamic pressure was 3.32 times as great 
as that against a plane, for P = 3.32 F, whereas theoretically the 
3.32 should be 4. .In this case, as in those preceding, the water in 
passing over the surface loses energy in friction and foam, so that 

* Ruhlman’s Hydromechanik (Hannover, 1879), P- 586. 


Experiments on Impulse and Reaction. Art. 153 403 


its velocity is diminished, and it should hence be expected that 
the experimental values of the dynamic pressures would be less 
than the theoretic values, as in general they are 
found to be. 

While the experiments here briefly described 
thoroughly confirm the results of theory, they 
further show it is the change in direction of the 
velocity when in contact with the surface which 
produces the dynamic pressure. If the stream 
strikes normally against a plane, the direction of its velocity is 
changed 90 °, and this is the same as the entire destruction of 
the velocity in its original direction, so that the dynamic pres¬ 
sure P should agree with the impulse F. This important princi¬ 
ple of change in direction will be theoretically exemplified later. 



Fig. 153 c>. 



The dynamic pressure which is produced by the direct reaction 
of a stream of water when issuing from a vertical orifice in the 

side of a vessel was measured by Ewart with 
the apparatus shown in Fig. 153 c, which will 
be readily understood without a detailed de¬ 
scription. The discussion of these experi¬ 
ments made by Weisbach * shows that the 
measured values of P were from 2 to 4 per¬ 
cent less than the theoretic value F as given 
by ( 153 ), so that in this case, also, theory and 
observation are in accordance. 



— -p 



o 


nn\ P 


Fig. 153 c. 


An experiment by Unwin, f illustrated in Fig. 153 d, is very 
interesting, as it perhaps explains more clearly than formula ( 152 ) 
why it is that the dynamic pressure 
due to impulse is double the static 
pressure. Two vessels having con¬ 
verging tubes of equal size were 
placed so that the jet from A was 
directed exactly into B. The head in 
A was kept uniform at 20^ inches, 

* Theoretical Mechanics, Coxe’s translation, vol. 1, p. 1004. 
f Encyclopedia Britannica, 9th Edition, vol. 12, p. 467. 

















































404 


Chap. 12 . Dynamic Pressure of Water 


when it was found that the water in B continued to rise until a 
head of 18 inches was reached. All the water admitted into A 
was thus lifted in B by the impulse of the jet, with a loss of 21- 
inches of head, which was caused by foam and friction. If 
such losses could be entirely avoided, the water in B might 
be raised to the same level as that in A. In the case shown in 
the figure where the water overflows from B, the impulse of the 
jet has not only to overcome the static pressure due to the 
head h , but also to furnish the dynamic pressure equivalent to 
a second head h in order to raise the water through that height. 
But the level in B can never rise higher than in A, for the 
velocity-head of the jet cannot be greater than that of the static 
head which generates it. 

Prob. 153 . Accepting as an experimental fact that the force of impulse 
or reaction is double the static pressure, show that the theoretic velocity 
of flow is V 2 gh. 

Art. 154 . Surfaces at Rest 

Let a jet of water whose cross-section is a impinge in perma¬ 
nent flow with the uniform velocity v upon a surface at rest. Let 
the surface be smooth, so that no resisting force of friction exists, 
and let the stream be prevented from spreading laterally. The 



water then passes over the surface, and leaves it with the original 
velocity v, producing upon it a dynamic pressure whose value 
depends upon its change of direction. At B in Fig. 154 a the 
stream is deflected normal to its original direction, and at D 
a complete reversal is effected. Let 0 be the angle between the 
initial and final directions, as shown. It is required to determine 
the dynamic pressure exerted upon the surface in the same direc¬ 
tion as that of the jet. In the above figures, as in those that followq 
the stream is supposed to lie in a horizontal plane, so that no 




























Surfaces at Rest. Art. 154 


405 


acceleration or retardation of its velocity will be produced by 
the action of gravity. 

The stream entering upon the surface exerts its impulse F 
in the same direction as that of its motion; leaving the surface, 


it exerts its reaction F in 
opposite direction to that 
of its motion. Let P be 
the dynamic pressure thus 
produced in the direction 
of the initial motion, F\ 
the component of the re¬ 
action F in the same direc¬ 
tion. Then 



i — cos 6 ) 


and inserting for F its value as given by ( 152 ), 

P =(i — cos 0 )W- ( 154 ), 

& 

which is the formula for the dynamic pressure in the direction 
of the impinging jet. If in this 6 = o°, the stream glides along 
the surface without changing its direction, and P becomes zero; 
if 6 is 90°, the resulting dynamic pressure is 

P=F=W- 

g 

and if 6 becomes 180°, a complete reversal of direction is obtained, 
and the resulting dynamic pressure that is exerted by the jet 
against the surface is 

P= 2 F = 2 W~ 

g 


These theoretic conclusions agree with the experimental results 
described in the last article. In the deduction of ( 154 )i the angle 
6 has been regarded as less than 90°, but the same formula results 
if 6 be considered greater than 90°, since then the sign of the 
reaction F\ is positive. 

The resultant dynamic pressure exerted upon the surface is 
found by combining by the parallelogram of forces the impulse F 









406 


Chap. 12 . Dynamic Pressure of Water 


and the equal reaction F. In Fig. 1546 it is seen that this resul¬ 
tant bisects the angle 180 — #, and that its value is 

P' = 2 F cos|(i8o — 6 ) = 2 sinf# • W- 

& 

It is usually, however, more important to ascertain the pressure 
in a given direction than the resultant. This can be found by 

taking the component of the resultant in that 
direction, or by taking the algebraic sum of 
the components of the initial impulse and the 
final reaction. 

To find the dynamic pressure P in a di¬ 
rection which makes an angle « with the 
entering and the angle 6 with the departing 
stream, the components in that direction are 

Pi = F cos« P 2 = — F cos# 
and the algebraic sum of these two components is 

P = F (cos** — cos#) = (cosa — cos#) W - ( 154) 2 

This becomes equal to F when a = o and # = 90°, as at B in 
Fig. 154 a, and also when « = 90° and # = 180°. When a = o° 
and # = 180 0 the entering and departing streams are parallel, as 
at D in Fig. 154 a, so that the value of P is 2 F, which in this case 
is the same as the resultant pressure. 

The formulas here deduced are entirely independent of the form 
of the surface, and of the angle with which the jet enters upon it. 
It is clear, however, if, as in the planes in Fig. 1 . 54 a, this angle is 
such as to allow shock to occur, that foam and changes in cross-sec¬ 
tion may result which will cause energy to be dissipated in heat. 
These losses will diminish the velocity v and decrease the theoretic 
dynamic pressure. These effects cannot be formulated, but it is a 
general principle, which is confirmed by experiment, that they may 
be largely avoided by allowing the jet to impinge tangentially upon 
the surface. 

In all the foregoing formulas the weight W which impinges upon 
the surface per second is the same as that which issues from the orifice 
or nozzle that supplies the stream, or 

W = wq = wav 



Fig. 154 c. 


Immersed Bodies. Art. 155 


407 


To find W it is hence necessary to use the methods of the preceding 
chapters to determine either the discharge q or the mean velocity v. 

Prob. 154 . If F is io pounds, a = 0 °, and 6 = 6o°, show that the pres¬ 
sure parallel to the direction of the jet is 5 pounds, that the pressure normal 
to that direction is 8.66 pounds, and that the resultant dynamic pressure is 
10 pounds. 


Art. 155 . Immersed Bodies 

When a body is immersed in a flowing stream, or when it is 
moved in still water, so that filaments are caused to change their 
direction, a dynamic pressure is exerted by the stream or overcome 



by the body. It is to be inferred from what has preceded that 
the dynamic pressure in the direction of the motion is proportional 
to the force of impulse of a stream which has a cross-section equal 
to that of the body, or 2 

P = m • wa~ 

in which m is a number depending upon the length and shape of 
the immersed portion, and whose value is 2 for a jet impinging 
normally upon a plane. 

Experiments made upon small plates held normally to the 
direction of the flow show that the value of m lies between 1.25 
and 1.75, the best determinations being near 1.4 and 1.5. It is 
to be expected that the dynamic pressure on a plate in a stream 
would be less than that due to the impulse of a jet of the same 
cross-section, as the filaments of water near the outer edges are 
crowded sideways in the latter case and hence do not impinge 
with full normal effect, and the above results confirm this sup¬ 
position. The few experiments on record were made with small 
plates, mostly less than 2 square feet in area, and they seem to 
indicate that the value of the coefficient m is greater for large 
surfaces than for small ones. 
























408 Chap. 12 . Dynamic Pressure of Water 

The determination of the dynamic pressure upon the end of 
an immersed cylinder or prism is difficult because of the resisting 
friction of the sides ; but it is well ascertained to be less than that 
upon a plane, of the same area, and within certain limits to de¬ 
crease with the length. For a conical or wedge-shaped body the 
dynamic pressure is less than that upon the cylinder, and it is 
found that its intensity is much modified by the shape of the rear 
surface of the body. 

When a body is so shaped as to gradually deflect the filaments 
of water in front, and to allow them to gradually close in again 
upon the rear, the impulse of the front filaments upon the body 
is balanced by the reaction of those in the rear, so that the resul¬ 
tant dynamic pressure is zero. The forms of boats and ships 
should be made so as to obtain this result as nearly as possible, 
and then the propelling force has only to overcome the frictional 
resistance of the surface upon the water. A body so shaped is 
said to have a “fair form” (Art. 183 ). 

The dynamic pressure produced by the impulse of ocean 
waves striking upon piers or lighthouses is often very great. 
The experiments of Stevenson on Skerryvore Island, where the 
waves probably acted with greater force than usual, showed that 
during the summer months the mean dynamic pressure per square 
foot was about 6oo pounds, and during the winter months about 
2100 pounds, the maximum observed value being 6100 pounds. 
At the Bell Rock lighthouse the greatest value observed was 
about 3000 pounds per square foot. The observations were made 
by allowing the waves to impinge upon a circular plate about 6 
inches in diameter, and the pressure produced was registered by 
' the compression of a spring. Such high unit-pressures do not 
probably act upon large areas of masonry which are exposed to 
wave action.* 

Prob. 155 . Compute the probable dynamic pressure upon a surface 
1 foot square when immersed in a current whose velocity is 9 feet per second, 
the direction of the current being normal to the surface. 

* Cooper on Ocean Waves, in Transactions American Society Civil 

Engineers, 1896, vol. 36, p. 150. 


Curved Pipes and Channels. Art. 156 


409 


Art. 156. Curved Pipes and Channels 

The dynamic pressures discussed in the preceding article 
have been those caused by. jets, or isolated streams, of water. 
There is now to be considered the case of dynamic pressures 
caused by streams flowing in pipes, conduits, or channels of any 
kind; these are sometimes called limited or bounded streams, 
the boundary being the surface whose cross-section is the wetted 
perimeter. When such a stream is straight and of uniform sec¬ 
tion, and all its filaments move with the same velocity v, the im¬ 
pulse, or the pressure which it can produce, is the quantity F 
given by the general expression in Art. 152 ; under these conditions 
it exerts no dynamic pressure, but if a body be immersed and 



held stationary, pressure is produced upon it. If its direction 
changes in an elbow or bend, pressure upon the bounding surface 
is produced; if its cross-section increases or decreases, pressure 
is developed or absorbed. 

The resultant dynamic pressure P' upon a curved pipe which 
runs full of water with the uniform velocity v depends upon the 
angle 6 between the initial and final directions, and must be the 
same as that produced upon a surface by an impinging jet which 
passes over it without change in velocity. The formula of Art. 
154 then directly applies, and 

P f = 2 sinj# • F = 2 sin|0 • W ~ 

if 0 = o°, there is no bend, and P' = o; if 0 = i8o°, the direc¬ 
tion of flow is reversed, and P' = 2 F. If the direction is twice 
reversed, as at C in Fig. 156#, the value of 0 is 360 , and the re- 
















410 


Chap. 12. Dynamic Pressure of Water 


sultant is the initial impulse F minus the final reaction F, or simply 
zero; in this case, however, there may be a couple which tends 
to twist the pipe, unless the impulse and reaction lie in the same 
straight line. 

The dynamic pressure developed in a unit of length of the curve 
will now be found. Let the pipe at A in Fig. 156a have the length 
SI, and let 0 be nearly o°, so that its value is the elementary angle 
SO. Then in the above formula P' becomes the elementary radial 
pressure SP h and 

SP X = 2 sin \ SO • F = F&O 


Now since 80 = Sl/R, where R is the radius of the curve, the dynamic 
pressure developed in the distance SI is FSl/R, and that for a unit of 
length is F/R. Accordingly, by Art. 153, this pressure is 


p _ F _ 2 wa v 2 
R~ R 2g 

The unit-pressure p' is found by dividing P x by a, and the correspond¬ 
ing head h A is found by dividing p f by w ; hence 



2 W V 2 

R 2 g 


and 


A v 2 

R2g 


are the values for one unit of length of the curve. The dynamic 
pressure-head hi is developed in every unit of length of the pipe. It 
is not known how these influence the static pressure or how they affect 
piezometers. Nor is it known whether they combine so that the dy¬ 
namic pressure becomes greater with the distance from the beginning 
of the curve. Undoubtedly, however, a part of h x is expended in 
causing cross-currents whereby impact results and some of the static 
head is lost. This loss should be proportional to h x and proportional 
to the length l of the curve, or, if d is the diameter of the pipe, 


/ m l v 2 d 

n = mi - = mi — 

R2g R 


1 Ti — y 

d 2g d 2g 


in which the curvature factor f x depends upon the ratio R/d. This 
investigation appears to indicate that pipes of the same diameter and 
of different curvatures give the same loss of head, if the central angle 
is the same; but, as seen in Art. 91, certain experiments seem to point 
to the conclusion that the loss per linear unit is greatest in the pipe 
having the longest radius. 



Curved Pipes and Channels. Art. 156 411 

The same reasoning applies approximately to the curves of 
conduits, canals, and rivers. In any length l there exists a radial 
dynamic pressure P h acting toward the outer bank and causing 
currents in that direction, which, in connection with the greater 
velocity that naturally there exists, tends to deepen the channel 
on that side. This may help to explain the reason why rivers run 
in winding courses. At first the curve may be slight, but the 
radial flow due to the dynamic 
pressure causes the outer bank 
to scour away; this increases 
the velocity v 2 and decreases V\ 

(Fig. 1565), and this in turn 
hastens the scour on the outer 
and allows material to be de¬ 
posited on the inner side. Thus 
the process continues until a 
state of permanency is reached, 
and then the existing forces tend to maintain the curve. The 
cross-currents which the radial pressure produces have been 
actually seen in experiments devised by Thomson,* and it is 
found that they move in the manner shown in the above figure, 
the motion toward the outer bank being in the upper part of the 
section, while along the wetted perimeter they flow toward the 
inner bank. When the slope is small and the mean velocity 
low, the influence of the cross-currents is relatively greater than 
for higher slopes, and this is probably one of the reasons why 
the sharpest curves are found in streams of slight slope. Per¬ 
haps another reason for this is that at very low velocities the law 
of flow is different, the head varying as the first power of the 
velocity (Art. 124). 

The elevation of the water on the outer bank of a bend is 
higher than on the inner. This is only a partial consequence of 
the radial dynamic pressure, as in straight streams it is also seen 
that the water surface is curved, the highest elevation being where 
the velocity is greatest. In this case cross-currents are also ob- 



* Proceedings Royal Society of London, 1878, p. 356. 












412 Chap. 12. Dynamic Pressure of Water 

served which move near the surface toward the center of the 
stream, and near the bottom toward the banks, their motion 
being due to the disturbance of the static pressure consequent 
upon the varying water level. 

Prob. 156. The mean velocity in a pipe is 9 feet per second. If it be 
laid on a curve of 3 feet radius, show that the dynamic pressure-head for 
each foot in length of the pipe is 0.84 feet. If the radius of the curve be 6 
feet, what is the dynamic pressure-head ? What is the dynamic pressure- 
head for each case when the mean velocity is 3 feet per second ? 

Art. 157. Water Hammer in Pipes 

When a valve in a pipe is closed while the water is flowing, 
the velocity of the water is retarded as the valve descends, and 
thus a dynamic pressure is produced. When the valve is closed 
quickly, this dynamic pressure may be much greater than that due 
to the static pressure, and it is then called “water hammer” or 
“water ram.” Pipes have often been known to burst under this 
cause, and hence the determination of the maximum dynamic 
pressure of the water hammer is a matter of importance. Fig. 
157a illustrates the phenomena of water hammer for the closing 


c 



of a valve at the end of a pipe where the water issues through a 
nozzle. At the entrance there is supposed to be a gage which 
registers the static unit-pressure pi while the flow is in progress, 
and the static unit-pressure p 0 when there is no flow. The ab¬ 
scissas represent time, and at B the valve begins to close. After 
a short interval of time BC the gage registers the unit-pressure 
Cc ; after another short interval the unit-pressure has decreased 













•Water Hammer in Pipes. Art. 157 


413 


to Dd , and a series of oscillations follows until finally the dis¬ 
turbance ceases. A diagram of this kind may be autographically 
drawn by suitable mechanism connected with the pressure gage, 
and such were made in the experiments conducted by Carpenter,* 
as also in those of Fletcher, f 

Let p represent the excess of maximum dynamic unit-pressure 
over the static unit-pressure when there is no flow; that is, the 
difference of the ordinates Cc and Ee. This is the excess unit- 
pressure due to the water hammer, and it is required to determine 
an expression for its value. It is first to be noted that the actual 
dynamic unit-pressure produced by the retardation of the veloc¬ 
ity is the difference of the ordinates Cc and Bb and this difference 
is p + po — pi. The dynamic pressure on the area a of the cross- 
section of the pipe is then (p + po — pi)a, and for brevity this 
may be represented by P. If this pressure be regarded as 
varying uniformly from o up to P during the time t in which the 
valve closes, its mean value is J P and its total impulse during 
this time is \ Pt. If l be the length of the pipe, w the weight of 
a cubic unit of water, and v the velocity during the flow, the total 
weight of water in the pipe is wal and its impulse is wal • v/g. 
Equating these expressions of the impulse there is found P = 2 
walv/gt , and replacing P by its value, there results 

_ v _j_ _ p Q (157)i 

gt 

as the excess dynamic unit-pressure due to closing the valve in 
the time t. This formula, having been deduced without consid¬ 
ering the fact that time is required for the transmission of stress 
through water, cannot be regarded as applicable to all cases. 

In Art. 5 it was shown that the velocity with which any dis¬ 
turbance is propagated through water is about 4670 feet per 
second, and this velocity may be represented by u. Now let the 
pipe of length / have an open valve at the end, and let the water 
be flowing through every section with the velocity v. Then the 

* Transactions American Society of Mechanical Engineers, 1894, vol. 15. 

f Engineering News, 1898, vol. 39, p. 323. 



414 Chap. 12. Dynamic Pressure of Water 

time l/u must elapse after the valve begins to close before the 
velocity begins to be checked at the upper end of the pipe, and 
the further time of l/u must elapse before the pressure due to this 
retardation can be transmitted back to the valve. The total 
time 2 l/u is then required before the gage at the valve can indi¬ 
cate the pressure due to the retardation of the velocity in the 
length /. Hence, if the time in which the valve closes be equal 
to or less than 2 //«, the time t in the above formula is to be re¬ 
placed by 2 l/u, and thus 

P=m v+Pl - Po (157), 

g 

is the maximum excess dynamic unit-pressure that can occur in 
the pipe. This depends upon the velocity of the water and upon 
the initial and final static pressures. 

The subject of water hammer in pipes is one of the most diffi¬ 
cult in hydromechanics, and the above investigation cannot be 
regarded as final. Formula (157) 1 is probably correct only for 
a certain law of valve closing. Formula (157) 2 , however, is cer¬ 
tainly correct, for it may be proved by other methods, one of 
which is as follows: When the water is in motion, the kinetic 
energy in a length Bl at the gage is waBl • v 2 / 2 g ; when it is brought 
to rest under the unit-stress S, its stress energy is aBl • S' 2 / 2 E, if 
E be the modulus of elasticity of the water.* Equating these 
expressions, and substituting p + p 0 — pi for 5, there results for 
the excess dynamic unit-pressure 

/> = (“) v + pi-po 

and this reduces to (157) 2 if E be replaced by wu 2 /g, which is 
its value according to formula (5). 

When v is in feet per second, and p 0 , p h and p are in pounds 
per square inch, these formulas become 

p = 0.027 (*/ t)v + pi-p 0 p = 63 v + pi- po (157)3 

the first of which is to be used when t is greater than 0 . 000428 / 
and the second when t is equal to or less than it, l being in feet. 


* Merriman’s Mechanics of Materials (New York, 1911), p. 306. 




Water Hammer in Pipes. Art. 157 


415 


From the first of these formulas the value of t, when p = o, is 

found to be j 

t = 0.027 —-— 
po ~ pi 

which is the time of valve closing in order that there may be no 
water hammer. For example, let p 0 be 83 and pi be 58 pounds 
per square inch, l be 1903 feet, and v be 5 feet per second, then t 
is 10.3 seconds. To prevent the effects of water hammer, it is 
customary to arrange valves so that they cannot be closed very 
quickly, and the last formula furnishes the means of estimating 
the time required in order that no excess of dynamic pressure 
over the static pressure p 0 may occur. 

The elaborate experiments of Joukowsky at Moscow in 1898 * 
have fully confirmed the truth of formula (157) 2 . Horizontal 
pipes of 2 , 4 , and 6 inches diameter, with lengths of 2494 , 1050 , 
and 1066 feet, were used, and the valve at the end was closed in 
0.03 seconds. Ten autographic recording gages were placed along 
the length of a pipe, and it was found that substantially the same 
dynamic pressure was produced at each, but that the time length 
of a wave was the shorter the farther the distance of a gage from 
the valve; this wave length is shown in the above figure by the 
distance BD. The following is a comparison of the observed 


For the 4-inch Pipe 


Velocity 

Observed 

Computed 

0-5 

31 

31 

1.9 

n 5 

Il8 

2.9 

168 

183 

4.1 

232 

258 

9.2 

5 i 9 

580 


For the 6-inch Pipe 


Velocity 

Observed 

Computed 

0.6 

43 

38 

1.9 

106 

118 

3-0 

173 

189 

5-6 

369 

353 

7-5 

426 

472 


values of p + p 0 — pi for a few of these experiments with the 
values computed from (157) 3 . It is seen that the observed are less 
than the computed values except in one instance, and Joukowsky 

* Stoss in Wasserleitungsrohren, St. Petersburg, 1900. Translation 
from the Memoirs of the St. Petersburg Academy of Sciences. 



















416 Chap. 12. Dynamic Pressure of Water 

concludes that, owing to the influence of the metal of the pipes, 
the velocity u with which stress is transmitted in the water is 
about 4200 instead of 4670 feet per second. This conclusion 
may be applied in practice by using 59 ^ instead of 63 ^ in (157) 3 . 

Fig. 157a shows the waves of pressure for a case where the 
valve is closed in a time greater than 2 l/u. Fig. 1575 shows 



the oscillations for two cases, the broken line being for t = 0.7 
seconds and the full line for t = 0.3 seconds, both cases referring 
to a pipe for which the time 2 l/u is about 0.6 seconds. It is seen 
that the crests of the waves are flat when the time of closing the 
valve is less than 2 l/u, and diagrams of this kind only were drawn 
in the experiments of Joukowsky. 

In computing the thickness of water pipes it is customary to 
add 100 pounds per square inch to the static pressure in order to 
allow for the influence of water hammer. This is equivalent, if 
pi is zero, to making po + 100 equivalent to ; when v is 3 feet 
per second, then p 0 is 89 pounds per square inch. Since these 
values of v and p are larger than the usual ones for a city water 
supply, the customary practice is on the safe side for this case, 
but it would not give sufficient security for the high velocities 
often used in pipe lines for power plants. When a wave of dy¬ 
namic pressure travels toward a dead end of a pipe, the water 
hammer at that end may be two or three times as great as the 
maximum pressure given by the formula. 

In the case of a water power plant supplied from a pipe or 
long penstock, a “surge tank” * may be placed near the lower end 
in order to prevent sudden changes in pressure due to sudden 

* Transactions American Society of Mechanical Engineers, 1908, p. 443. 










Moving Vanes. Art. 158 


417 


changes in load on the wheels and the consequent fluctuations of 
velocity within the feeding pipe. 

Prob. 157 . The pressure-head at the entrance to a nozzle is 400 feet 
when there is no flow and 200 feet when the water is flowing. The pipe is 
1500 feet long and the velocity in it is 4 feet per second when the nozzle 
is in operation. Compute the excess dynamic pressure when the valve is 
closed in 0.7 seconds and also when it is closed in 0.3 seconds. 

Art. 158. Moving Vanes 

A vane is a plane or curved surface which moves in a given 
direction under the dynamic pressure exerted by an impinging 
jet or stream. The direction of the motion of the vane depends 
upon the conditions of its construction; for example, the vanes 
of a water wheel can only move in a circumference around its axis. 
The simplest case for consideration, however, is that where the 
motion is in a straight line, and this alone will be considered in 
this article. The plane of the stream and vane is to be taken as 
horizontal, so that no direct action of gravity can influence the 
action of the jet. 

Let a jet with the velocity v impinge upon a vane which moves 
in the same direction with the velocity u, and let the velocity of 
the jet relative to the surface at the point 
of exit make an angle ft with the reverse 
direction of u , as shown in Fig. 158a. 

The velocity of the stream relative to the 
surface is v — u , and the dynamic pres¬ 
sure is the same as if the surface w r ere at 
rest and the stream moving with the ab¬ 
solute velocity v — u. Hence formula (154) 1 directly applies, 
replacing v by v - u and 6 by 180 0 - ft, and the dynamic pres¬ 
sure is 

P = ( 1 + cos/3) 

g 

In this formula W is not the weight of the water which issues from 
the nozzle, but that which strikes and leaves the vane, or IV = wa 
(v - u) ; for under the condition here supposed the vane moves 








418 


Chap. 12. Dynamic Pressure of Water 


continually away from the nozzle, and hence does not receive 
all the water which it delivers. 

Another method of deducing the last equation is as follows: 
At the point of exit let lines be drawn representing the velocities 
v — u and u ; then, completing the parallelogram, the line V\ 
is the absolute velocity of the departing jet (Art. 28). Let 6 be 
the angle which Vi makes with the direction of u , and /3 as before 
the angle between v — u and the reverse direction of a. Then 
the dynamic pressure on the vane is that due to the absolute 
impulse of the entering and departing streams: the former of 
these is W • v/g and the latter is W • V\ cos 6/g. Hence the result¬ 
ant dynamic pressure in the direction of the motion of the vane 
is the difference of these impulses, or 

p = \y v 

g 

But from the triangle between Vi and u 

Vi cos 6 =u — (y — u) cos/3 

Inserting this, the value of the dynamic pressure is 

P = (i + cos/3) PL -—— 

g 

which is the same as that found before. If /3 = i8o°, there is no 
pressure, and if /3 = o°, the stream is completely reversed, and 
P attains its maximum value. The dynamic pressure may be 
exerted with different intensities upon different parts of the vane, 
but its total value in the direction of the motion is that given 
by the formula. 

Usually the direction of the motion is not the same as that of 
the jet. This case is shown in Fig. 1586, where the arrow marked 
F designates the direction of the impinging jet, and that marked 
P the direction of the motion of the vane, a being the angle be¬ 
tween them. The jet having the velocity v impinges upon the 
vane at A, and in passing over it exerts a dynamic pressure P 
which causes it to move with the velocity u. At A let lines be 
drawn representing the intensities and directions of v and u, and 
let the parallelogram of velocities be formed as shown; the line 




Moving Vanes. Art. 158 


419 


V then represents the velocity of the water relative to the vane 
at A. The stream passes over the surface and leaves it at B 
with the same relative velocity V, if not retarded by friction or 
foam. At B let lines be drawn 
to represent u and V, and let 
ft be the angle which V makes 
with the reverse direction of u ; 
let the parallelogram be com¬ 
pleted, giving v\ for the abso¬ 
lute velocity of the departing 
water, and let 6 be the angle 
which it makes with u. The 
total pressure in the direction 
of the motion is now to be regarded as that caused by the com¬ 
ponents in that direction of the initial and the final impulse of 
the water. The impulse of the stream before striking the vane 
is W • v/g and its component in the direction of the motion is 
W • v cos a/g. The impulse of the stream as it leaves the vane 
is W • Vi/g and its component in the direction of the motion is 
W • V\ cos Q/g. The difference of these components is the result¬ 
ant dynamic pressure in the given direction, or 

p = pp ^ cos« z'i cos 0 (To8) 

g 

This is a general formula for the dynamic pressure in any given 
direction upon a vane moving in a straight line, if a and 0 be the 
angles between that direction and those of v and v\. If the surface 
be at rest, v and v\ are equal and the formula reduces to (154) 2 . 

If it be required to find the numerical value of P, the given 
data are the velocities v and u and the angles « and ft. The term 
Vi cos 6 is hence to be expressed in terms of these quantities. From 
the triangle at B between V\ and u , there is found 

Vi cos 0 — u — V cos/3 

and substituting this, the formula becomes 

p — j y v cosu ~ ^ 4- F cos ft 

g 



Fig. 1586 . 












420 


Chap. 12. Dynamic Pressure of Water 


which is often a more convenient form for discussion. The 
value of V is found from the triangle at A between u and v, 

^ us * V 2 = u 2 + v 2 — 2 UV cos a 

and hence the dynamic pressure P is fully determined in terms of 
the given data. 

In order that the stream may enter tangentially upon the vane, 
and thus prevent foam, the curve of the vane at A should be tan¬ 
gent to the direction of V. This direction can be found by ex¬ 
pressing the angle <f> in terms of the given angle «. Thus from 
the relation between the sides and angles of the triangle included 
between u, v, and V there is found 

sin (<f> — a) / sin<£ = u/v 

which is easily reduced to the form 

cot <f) = cota- t— 

v sm« 

from which </> can be computed when u, v, and « are given. 
For example, if u be equal to and if a be 30 °, then cot</> is 0 . 732 , 
whence the angle <f> should be 53 !°, in order that the jet may enter 
without impact. If the angle made by the vane with the direc¬ 
tion of motion be greater or less than this value, some loss due to 
impact will result at the given speed. 

Prob. 158 . Given u = 86.6 and v = 100.0 feet per second, and a = 30°. 
What should be the value of the angle if/ in order that no loss by impact 
may occur ? Draw the parallelogram showing the velocities u, v, and V. 

Art. 159. Work derived from Moving Vanes 

The work imparted to a moving vane by the energy of the 
impinging water is equal to the product of the dynamic pressure 
P, which is exerted in the direction of the motion and the space 
through which it moves. If u be the space described in one 
second, or the velocity of the vane, the work per second is 

k = Pu 

This expression is now to be discussed in order to determine the 
value of u which makes k a maximum. 



Work derived from Moving Vanes. Art. 159 


421 


When the vane moves in a straight line in the same direction 
as the impinging jet and the water enters it tangentially, as shown 
in Fig. 1546, the work imparted is found by inserting for P its 
value from (154)i. If a be the area of the cross-section of the jet 
and w the weight of a cubic unit of water, the weight W is wa 
(v — u), and then 

k = (i + cos/3) W — —= (i + cos/3) wa —— 11 

g g 

The value of u which renders k a maximum is obtained by 
equating to zero the derivative of k with respect to u , or 

~ = (i + cos /3) — ( v 2 — 4 Z m + 3 W 2 ) = o 

g 

from which the value of u is f v , and accordingly 


k = 2 t(i + cos^) wa 


V 3 

2 g 


is the maximum work that can be done by the vane in one 
second. The theoretic energy of the impinging jet is 


K 


W— = wa — 


2 g 


2 g 


and the efficiency of the vane is the ratio of the effective work 
of the vane to the theoretic energy of the water, or 

e = k/K = 2 V (1 + cos/3) 

If 0 = 180 0 , the jet glides along the vane without producing work 
and e = o; if /3 = 90 °, the water departs from the vane normal 
to its original direction and e = /y; if /3 = o°, the direction of the 
stream is reversed and e = Jf. 


It appears from the above that the greatest efficiency which can 
be obtained by a vane moving in a straight line under the impulse 
of a jet of water is ; that is, the effective work is only about 59 per¬ 
cent of the theoretic energy attainable. This is due to two causes: 
first, the quantity of water which reaches and leaves the vane per second 
is less than that furnished by the nozzle or mouthpiece from which the 
water issues; and, secondly, the water leaving the vane still has an 
absolute velocity of %v. A vane moving in a straight line is therefore 
a poor arrangement for utilizing energy, and it will also be seen upon 







422 


Chap. 12. Dynamic Pressure of Water 


reflection that it would be impossible to construct a motor in which a 
vane would move continually in the same direction away from a fixed 
nozzle. The above discussion therefore gives but a rude approxima¬ 
tion to the results obtainable under practical conditions. It shows 
truly, however, that the efficiency of a jet which is deflected normally 
from its path is but one-half of that obtainable when a complete 
reversal of direction is made. 

v 

Water wheels which act under the impulse of a jet consist of 
a series of vanes arranged around a circumference which by the 
motion are brought in succession before the jet. In this case the 
quantity of water which leaves the wheel per second is the same 
as that which enters it, so that W does not depend on the velocity 
of the vanes, as in the preceding case, but is a constant whose 
value is wq, where q is the quantity furnished per second. A close 
estimate of the efficiency of a series of such vanes can be made 
by considering a single vane and taking W as a constant. The 
water is supposed to impinge tangentially and the vane to move 
in the same line of direction as the jet (Fig. 158u). Then the 
work which is imparted in one second by the water to the mov¬ 
ing vane is , * 

k = (i+ cos/3) W ~ u - 

g 

This becomes zero when u = o or when u = v, and it is a maxi¬ 
mum when u = %v, or when the vane moves with one-half the 
velocity of the jet. Inserting this value of u, 

k = \ (i T- cos/3) W — 

and, dividing this by the theoretic energy of the jet, the effi¬ 
ciency of the vane is found to be 

e = \ (i 4- cos/3) 

When /3 = i8o°, the jet merely glides along the surface without 
performing work and e = o; when /3 = qo°, the jet is deflected 
normally to the direction of the motion and e — b ; when /3 = o°, 
a complete reversal of direction is obtained and the efficiency 
attains its maximum value e = i. 



Revolving Vanes. Art. 160 


423 


These conclusions apply closely to the vanes of a water wheel 
which are so shaped that the water enters upon them tangentially 
in the direction of the motion. If the vanes are plane radial 
surfaces, as in simple paddle wheels, the water passes away nor¬ 
mally to the circumference, and the highest obtainable efficiency 
is about 50 percent. If the vanes are curved backward, the effi¬ 
ciency becomes greater, and, neglecting losses in impact and fric¬ 
tion, it might be made nearly unity, and the entire energy of the 
stream be realized, if the water could both enter and leave the 
vanes in a direction tangential to the circumference. The in¬ 
vestigation shows that this is due to the fact that the water leaves 
the vanes without velocity; for, as the advantageous velocity 
of the vane is \v, the water upon its surface has the relative 
velocity v — = \v ; then, if /3 — o°, its absolute velocity as 

it leaves the vane is = o. If the velocity of the vanes 

is less or greater than half the velocity of the jet, the efficiency 
is lessened, although slight variations from the advantageous 
velocity do not practically influence the value of e. 

Prob 159 . A nozzle 0.125 feet in diameter, whose coefficient of dis¬ 
charge is 0.95, delivers water under a head of 82 feet against a series of small 
vanes on a circumference whose diameter is 18.5 feet. Find the most ad¬ 
vantageous velocity of revolution of the circumference. 


Art. 160. Revolving Vanes 

When vanes are attached to an axis around which they move, 
as is the case in water wheels, the dynamic pressure which is 
effective in causing the motion is that tangential to the circum¬ 
ferences of revolution; or at any given point this effective pres¬ 
sure is normal to a radius drawn from the point to the axis. In 
Fig. 160 are shown two cases of a rotating vane; in the first the 
water passes outward or away from the axis, and in the second it 
passes inward or toward the axis. The reasoning, however, is 
general and will apply to both cases. At A, where the jet enters 
upon the vane, let v be its absolute velocity, V its velocity rela¬ 
tive to the vane, and u the velocity of the point A ; draw u normal 
to the radius r and construct the parallelogram of velocities as 


424 


Chap. 12. Dynamic Pressure of Water 


shown, a being the angle between the directions of u and v, and 
<f> that between u and V. At B, where the water leaves the vane, 
let Ui be the velocity of that point normal to the radius r h and Vi 
the velocity of the water relative to the vane; then constructing 



Fig. 160 . 


the parallelogram, the resultant of U\ and V\ is Vi, the absolute 
velocity of the departing water. Let ft be the angle between V\ 
and the reverse direction of ii\, and 6 be the angle between the 
directions of Vi and U\. 

The total dynamic pressure exerted in the direction of the 

motion will depend upon the values of the impulse of the entering 

and departing streams. The absolute impulse of the water before 

entering is W • v/g, and that of the water after leaving is W • Vi/g , 

Let the components of these in the directions of the motion of 

the vane at entrance and departure be designated by P and Pi ; 

then . 

p = \y ^ cos** p __ jy V\ cos 6 

s 1 g 

These, however, cannot be subtracted to give the resultant dy¬ 
namic pressure, as was done in the case of motion in a straight 
line, because their directions are not parallel, and the velocities 
of their points of application are not equal. The resultant dy¬ 
namic pressure is not important in cases of this kind, but the 
above values will prove useful in the next article in investigating 
the work that can be delivered by the vane. 





Revolving Vanes. Art. 160 


425 


If n be the number of revolutions around the axis in one sec¬ 
ond, the velocities u and u\ are 

u— 27 rrn U\ = 2irY\n 

and accordingly the relation obtains 

iii/ii = ri/r or u x y = uy x 

which shows that the velocities of the points of entrance and exit 
are directly proportional to their distances from the axis. If r 
and Y\ are infinity, u equals u x and the case is that of motion in 
a straight line as discussed in Art. 158. 

The relative velocities V x and V are connected with the veloc¬ 
ities of rotation u x and u by a simple relation. To deduce it, 
imagine an observer standing on the outward-flow vane and 
moving with it; he sees a particle of weight w a.t A which to him 
appears to have the velocity V, while the same particle at B 
appears to have the velocity V x ; the difference of their kinetic 
energies, or w(V 2 — V 2 ) / 2 g, is the apparent gain of the wheel- 
energy. Again, consider an observer standing on the earth and 
looking down upon the vane; from his point of view the energy 
gained is w(u x 2 — u 2 )/ 2 g. Now these two expressions for the , 
gain of the wheel in energy must be equal, or 

Vi 2 —V 2 = ui 2 — u 2 (160) 

and this is the formula by which V x is to be computed when V 
and the velocities of rotation are known. The same reasoning 
applies to the inward-flow vane by using the word “ loss ” instead 
of “ gain,” and the same formula results. 

The given data for a revolving vane are the angles and ft, 
the radii y and r x , the velocity v, the number of revolutions per 
second, and the weight of water delivered to the vane per second. 
The value of v cos a, and hence that of Pi, is immediately known. 
From the speed of revolution the velocities u and u x are found. 
The relative velocity V is, from the triangle between u and v, 

V = v sin«/sin 4> 

and by (160) the relative velocity \\ is then found from 

Vi = u 2 - u 2 + V 2 


426 Chap. 12. Dynamic Pressure of Water 

Lastly, the value of V\ cos<9, from the triangle between U\ and 
^ * s v\ cos# = u\ — Vi cos/3 

and accordingly the values of the dynamic pressures P and Pi are 
fully determined. Numerical values of these, however, are never 
needed, but the work due to them is of much importance, as will 
be explained in the next article. 

Prob. 160 . Given r — 2 feet ,*'1=3 feet, a = 45 0 , if/ = go°, v = 100 
feet per second, and n = 6 revolutions per second. Compute the velocities 
u, Mi, V, and V\. 


Art. 161. Work derived from Revolving Vanes 

The investigation in Art. 159 on the work and efficiency of 
a revolving vane supposes that all its points move with the same 
velocity, and that the water enters upon it in the same direction 
as that of its motion, or that a = o. This cannot in general be 
the case in water motors, as then the jet would be tangential to 
the circumference and no water could enter. To consider the 
subject further the reasoning of the last article will be continued, 
and, using the same notation, it will be plain that the work of a 
series of vanes arranged around a wheel may be regarded as that 
due to the impulse of the entering stream in the direction of the 
motion around the axis minus that due to the impulse of the de¬ 
parting stream in the same direction, or 

k = Pu — P\U\ 

Here P and Pi are the pressures due to the impulse at A and 
B (Fig. 160), and inserting their values as found, 

yt. _ ]y UV COS** — U \ V \ cos 0 (161) 

g 1 

This is a general formula applicable to the work of all wheels of 
outward or inward flow, and it is seen that the useful work k 
consists of two parts, one due to the entering and the other to 
the departing stream. 

Another general expression for the work of a series of vanes 
may be established as follows : Let v and Vi be the absolute veloc- 



Work derived from Revolving Vanes. Art. 161 427 

ities of the entering and departing water; the theoretic energy 
of this water is W • v 2 / 2 g, and when it leaves the wheel it still 
has the energy W • Vi 2 / 2 g. Neglecting losses of energy in impact 
and friction the work that can be derived from the wheel is 

k = W —^ (161)2 

This is a formula of equal generality with the preceding, and like 
it is applicable to all cases of the conversion of energy into work 
by means of impulse or reaction. In both formulas, however, 
the plane of the vane is supposed to be horizontal, so that no fall 
occurs between the points of entrance and exit. 

Formula (160) may be demonstrated in another way by 
equating the values of k in the preceding formulas; thus 

uv cosot — U]Vi cos 6 = \ {v 2 — v 2 ) 

Now from the triangle at A between u and v 

v 2 = V 2 — u 2 + 2 uv cos a 

and from the triangle at B between U\ and Vi 

V 2 = V 2 — U 2 + 2 U\V\ COS 0 

Inserting these values of v 2 and v 2 the equation reduces to 

V 2 — V 2 = uA - u 2 

This shows that if U\ be greater than u, as in the outward-flow 
vane, then Vi is greater than V ; if U\ is less than u, as in an in¬ 
ward-flow vane, then V\ is less than V. 



The above principles will now be applied to the simple case 
of an outward-flow wheel driven by a fixed nozzle, as in Fig. 1()T?. 







428 


Chap. 12. Dynamic Pressure of Water 


The wheel is so built that r — 2 feet, r 1 = 3 feet, a = 45 0 , </> = 90 °, 
and ft = 30 °. The velocity of the water issuing from the nozzle 
is v = 100 feet per second, and the discharge per second is 2.2 
cubic feet. It is required to find the work of the wheel and the 
efficiency when its speed is 337.5 revolutions per minute. 

The theoretic work of the stream per second is the weight 
delivered per second multiplied by its velocity-head, or 

k = 62.5 X 2.2 X 0.01555 X 100 2 = 21 380 foot-pounds 

which gives 38.9 theoretic horse-powers. The actual work of the 
wheel, neglecting losses in foam and friction, can be computed 
either from (161)i or (161) 2 . In order to use the first of these, 
however, the velocities u, u x , v x , and the angle 6 must be found, 
and to use the second, v x must be found; in each case V and IT 
must be determined. 

The velocities u and u,\ are found from the given speed of 
5.625 revolutions per second, thus : 

u =2 X 3.1416 X 2 X 5.625 = 70.71 feet per second; 

U\ = ij X 70.71 = 106.06 feet per second. 

The relative velocity V at the point of entrance is found from 
the triangle between V and v, which in this case is right-angled; 

V = v cos(</> — «) = v cos 45 0 = 70.71 feet per second. 

The relative velocity IT at the point of exit is found from the 
relation (160), which gives IT = u x = 106.06 feet per second. 
And since u x and V x are equal, v\ bisects the angle between IT 
and «i, and accordingly 

$ = i ( 180 0 — /5) = 75 degrees. 

The value of the absolute velocity v x then is 

Vi = 2 Mi cos# = 54.90 feet per second, 
and V\ 2 / 2 g is the velocity-head lost in the escaping water. 

The work of the wheel per second, computed either from (161)i 
or (161) 2 , is now found to be k = 14 934 foot-pounds or 27.2 
horse-powers, and hence the efficiency, or the ratio of this work 
to the theoretic work, is e = 0 . 699 . Thus 30.1 percent of the 


Revolving Tubes. Art. 162 


429 


energy of the water is lost, owing to the fact that the water leaves 
the wheel with such a large absolute velocity. 

In this example the speed given, 337.5 revolutions per minute, 
is such that the direction of the relative velocity V is tangent to the 
vane at the point of entrance. For any other speed this will not be 
the case, and thus w T ork will be lost in shock and foam. It is observed 
also that the approach angle a is one-half of the entrance angle <f >; 
with this arrangement the velocities u and V are equal, as also u x and 
Had the angle (3 been made smaller the efficiency of the wheel 
w r ould have been higher. 

Prob. 161 . Compute the power and efficiency for the above example 
if the angle (3 be 15 0 instead of 30°. Explain why /3 cannot be made very 
small. 


Art. 162. Revolving Tubes 


The water which glides over a vane can never be under static 
pressure, but when two vanes are placed near together and con¬ 
nected so as to form a closed tube, there may exist in it static 
pressure if the tube is filled. This is the condition in turbine 
wheels, where a number of such tubes, or buckets, are placed 
around an axis and water is forced through 
them by the static pressure of a head. The 
work in this case is done by the dynamic 
pressure exactly as in vanes, but the existence 
of the static pressure renders the investiga¬ 
tion more difficult. 

The simplest instance of a revolving tube 
is that of an arm attached to a vessel rotat¬ 
ing about a vertical axis, as in Fig. 162. It 
was shown in Art. 29 that the water surface in 
this case assumes the form of a paraboloid, 
and if no discharge occurs, it is clear that the 
static pressures at any two points B and A 
are measured by the pressure-heads Hi and H reckoned upwards 
to the parabolic curve, and, if the velocities of those points are 



Fig. 162. 


U\ and u : that 


Hi 


u 1 
2 g 


u 


= H — — = h 
2 g 










430 


Chap. 12. Dynamic Pressure of Water 


Now suppose an orifice to be opened in the end of the tube and the 
flow to occur, while at the same time the revolution is continued. 
The velocities V\ and V diminish the pressure-heads so that the 
piezometric line is no longer the parabola, but some curve repre¬ 
sented by the lower broken line in the figure. Then, according 
to the theorem of Art. 31, that pressure-head plus velocity-head 
remains constant during steady flow, if no loss of energy occurs, 

#! + — = H + — = h (162) 

2 g 2 g 2 g 2 g 


in which Hi and H are the heads due to the actual static pressures. 
This is the theorem which gives the relation between pressure- 
head, velocity-head, and rotation-head at any point of a revolving 
tube or bucket. If the tube is only partly full, so that the flow 
occurs along one side, like that of a stream upon a vane, then there 
is no static pressure, and the formula becomes the same as (160). 


An apparatus like Fig. 162, but having a number of arms from 
which the flow issues, is called a reaction wheel, since the dynamic 
pressure which causes the revolution is wholly due to the reaction 
of the issuing water. To investigate it, the general formula (161)x 
may be used. Making u = o, the work done upon the wheel by 
the water is 


— t nr ~ u i^i cos # _ \y w i cos/Q — u 2 


k = W 


g 


g 


But since there is no static pressure at the point B, the value 
of Vi is, from (162), or also from Art. 29, 

Vi = 2 gh + u 2 

The work that can be derived from the wheel now is 


£ _ jy Ui CO ft V 2 g/z+^i 2 — Ui 2 

g 

This becomes nothing when u\ = o, or when iii 2 = 2 gh cot 2 /3, 
and by equating the first derivative to zero it is found that k 
becomes a maximum when the velocity is given by 













Revolving Tubes. Art. 162 


431 


Inserting this advantageous velocity, the maximum work is 

k = Wh(i — sin/3) 

and therefore the efficiency of the reaction wheel is 

e — i — sin/3 

When /3 = 90 °, both U\ and e become o, for then the direction of 
the stream is normal to the circumference and no reaction can 
occur in the direction of revolution. When /3 = o, the efficiency 
becomes unity, but the velocity U\ becomes infinity. In the 
reaction wheel, therefore, high efficiency can only be secured by 
making the direction of the issuing water directly opposite to that 
of the revolution, and by having the speed very great. If (3 = 
i 9°.5 or sin /3 = J, the advantageous velocity U\ becomes V 2 gh 
and e becomes 0 . 67 . The effect of friction of the water on the 
sides of the revolving tube is not here considered, but this will 
be done in Art. 172. 

Prob. 162 a. Compute the theoretic efficiency of the reaction wheel 

when 0 = 180°, /3 = o°, and u\ = V2 gh. 

* 

Prob. 162 b. A reaction wheel has /3 = 30°, n = 0.302 meters, and h = 
4.5 meters. Compute the most advantageous number of revolutions per 
minute. If the quantity of water delivered to the wheel is 1600 liters per 
minute, compute the power of the wheel in metric horse-powers and in kilo¬ 
watts. 

Prob. 162 c. When l is in meters, v in meters per second, and p, pi, 
and po are in kilograms per square centimeter, the formulas ( 157 )« for water 
hammer become 

p = 0.0204 (///) v 4 - Pi — Po P = I 4-5 v + Pi ~ Po 
the first of which is to be used when t is greater than 0.001404/ and the second 
when t is equal to or less than it, l being in meters. 



432 


Chap. 13. Water Wheels 


CHAPTER 13 
WATER WHEELS 

Art. 163. Conditions of High Efficiency 

A hydraulic motor is an apparatus for utilizing the energy 
of a waterfall. It generally consists of a wheel which is caused to 
revolve either by the weight of water falling from a higher to a 
lower level, or by the dynamic pressure due to the change in direc¬ 
tion and velocity of a moving stream. When the water enters 
at only one part of the circumference, the apparatus is called a 
water wheel; when it enters around the entire circumference, it 
is called a turbine. In this chapter and the next these two classes 
of motors will be discussed in order to determine the conditions 
which render them most efficient. Overshot wheels, which move 
under the weight of water caught in their buckets, and undershot 
wheels, which move under the impact of a flowing stream, are 
forms that have been used for many centuries. Impulse wheels, 
which owe their motion to a jet of water striking their vanes 
with high velocity, were perfected in the nineteenth century. 

The efficiency e of a motor ought, if possible, to be independent 
of the amount of water used, or if not, it should be the greatest 
when the water supply is low. This is very difficult to attain. 
It should be noted, however, that it is not the mere variation in 
the quantity of water which causes the efficiency to vary, but it 
is the losses of head which are consequent thereon. For instance, 
when water is low, gates must be lowered to diminish the area of 
orifices, and this produces sudden changes of section which 
diminish the effective head h. A complete theoretic expression 
for the efficiency will hence not include W, the weight of water 
supplied per second, but it should, if possible, include the losses 
of energy or head which result when W varies. The actual effi¬ 
ciency of a motor can only be determined by tests with the fric- 






Conditions of High Efficiency. Art. 163 


433 


tion brake (Art. 149); the theoretic efficiency, as deduced from 
formulas like those of the last chapter, will as a rule be higher 
than the actual, because it is impossible to formulate accurately 
all the sources of loss. Nevertheless the deduction and discus¬ 
sion of formulas for theoretic efficiency are very important for the 
correct understanding and successful construction of all kinds 
of hydraulic motors. 

W hen a weight of water W falls in each second through the 
height h y or when it is delivered with the velocity v, its theoretic 
energy per second is 



K = IV h 


or 


The actual work per second equals the theoretic energy minus 
all the losses of energy. These losses may be divided into two 
classes: first, those caused by the transformation of energy into 
heat; and second, those due to the velocity V\ with which the 
water reaches the level of the tail race. The first class includes 
losses in friction, losses in foam and eddies consequent upon sud¬ 
den changes in cross-section or from allowing the entering water 
to dash improperly against surfaces; let the loss of work due to 
this be Wh'y in which h! is the head lost by these causes. The 
second loss is due merely to the fact that the departing water 
carries away the energy W • V\ 2 / 2 g. The work per second im¬ 
parted by the water to the wheel then is 



and dividing this by the theoretic energy the efficiency is, 



(163) 


in which v is the velocity due to the head h. This formula, al¬ 
though very general, must be the basis of all discussions on the 
theory of water wheels and motors. It shows that e can only 
become unity when h f = o and V\ = o, and accordingly the two 
following fundamental conditions must be fulfilled in order to 
secure high efficiency: 



434 


Chap. 13. Water Wheels 


1. The water must enter and pass through the wheel without 

losing energy in friction and foam. 

2. The water must reach the level of the tail race without ab¬ 

solute velocity. 

These two requirements are expressed in popular language by the 
well-known maxim “the water should enter the wheel without 
shock and leave without velocity.” Here the word ‘ ‘ shock ” means 
that method of introducing the water upon the wheel which 
produces foam and eddies. 

The friction of the wheel upon its bearings is included in the lost 
work when the power and efficiency are actually measured as described 
in Art. 149. But as this is not a hydraulic loss it should not be in¬ 
cluded in the lost work k f when discussing the wheel merely as a user 
of water, as will be done in this chapter. The amount lost in shaft 
and journal friction in good constructions may be estimated at 2 
or 3 percent of the theoretic energy, so that in discussing the hydrau¬ 
lic losses the maximum value of e will not be unity, but about 0.98 
or 0.97. This will usually be rendered considerably smaller by the 
friction of the wheel upon the air or water in which it moves, and 
which will here not be regarded. The efficiency given by (163) is 
called the hydraulic efficiency to distinguish it from the actual efficiency 
as determined by the friction brake. 

Prob. 163 . A wheel using 70 cubic feet of water per minute under a head 
of 12.4 feet has an efficiency of 63 percent. What effective horse-power 
does it deliver ? 


Art. 164. Overshot Wheels 

In the overshot wheel the water acts largely by its weight. 
Idgure 164 shows an end view or vertical section, which so fully 
illustrates its action that no detailed explanation is necessary. 
The total fall from the surface of the water in the head race or 
flume to the surface in the tail race is called /?, and the weight of 
water delivered per second to the wheel is called W. Then the 
theoretic energy per second imparted to the wheel is Wh. It is 
required to determine the conditions which will render the effec¬ 
tive work of the wheel as near to Wh as possible. 

The total fall may be divided into three parts : that in which 
the water is filling the buckets, that in which the water is descend- 


Overshot Wheels. Art. 164 


435 


ing in the filled buckets, and that which remains after the buckets 
are emptied. Let the first of these parts be called h 0 , and the 
last hi. In falling the dis¬ 
tance h 0 the water acquires 
a velocity v 0 which is approxi¬ 
mately equal to V 2gh 0 , and 
then, striking the buckets, 
this is reduced to u, the tan¬ 
gential velocity of the wheel, 
whereby a loss of energy in 
impact occurs. It then de¬ 
scends through the distance 
h — ho — h\, acting by its 
weight alone, and finally, 
dropping out of the buckets, 
reaches the level of the tail 
race with a velocity which 
causes a second loss of energy. Let li' be the head lost in enter¬ 
ing the buckets, and let v\ be the velocity of the water as it reaches 
the level of the tail race. Then the hydraulic efficiency of the 
wheel is given by the general formula (163), or 

e = i — — — 

h v 2 

and to apply it, the values of h! and Vi are to be found. In this 
equation v is the velocity due to the head h, or v = V 2 gh. 

The head lost in impact when a stream of water with the 
velocity Vo is enlarged in section so as to have the smaller velocity 
u, is, as proved in Art. 76, 

y _ (v 0 — U) 2 _ Vp 2 — 2 v 0 u + u 2 

2 g 2 g 

The velocity V\ with which the water reaches the tail race depends 
upon the velocity u and the height h\. Its kinetic energy as it 
leaves the buckets is W • u 2 / 2 g, the potential energy of the fall 
Jh is Wh\, and the resultant kinetic energy as it reaches the tail 
race is W • v 2 / 2 g ; hence the value of v\ is 

Vi = Vw 2 + 2 gh\ 





























436 


Chap. 13. Water Wheels 


e — i 


Inserting these values of h' and Vi in the formula for e, and 
placing for v 2 its equivalent 2 gh, there is found 

Vp 2 — 2VqII + 2 U 2 ~\~ 2g1l\ 

2 gh 

The value of u which renders e a maximum is found by equat¬ 
ing the first derivative to zero, which gives 

u = 

or the velocity of the wheel should be one-half that of the entering 
water. Inserting this value, the hydraulic efficiency correspond¬ 
ing to the advantageous velocity is 

W + wh 

2 gh 

and lastly, replacing Vo 2 by its value 2 gh 0 , it becomes 

1 h 0 hi 

2 h h 


(164) 


which is the maximum efficiency of the overshot wheel. 

This investigation shows that one-half of the entrance fall 
ho and the whole of the exit fall h\ are lost, and it is hence plain 
that in order to make e as large as possible both // 0 and hi should 
be as small as possible. The fall h 0 is made small by making the 
radius of the wheel large; but it cannot be made zero, for then 
no water would enter the wheel ; it is generally taken so as to 
make the angle # 0 about 10 or 15 degrees. The fall hi is made 
small by giving to the buckets a form which will retain the 
water as long as possible. As the water really leaves the wheel 
at several points along the lower circumference, the value of hi 
cannot usually be determined with exactness. 

The practical advantageous velocity of the overshot wheel, as 
determined by the method of Art. 149, is found to be about 0 . 4 ^ 0 , 
and its efficiency is found to be high, ranging from 70 to 90 percent. 
In times of drought, when the water supply is low, and it is desirable 
to utilize all the power available, its efficiency is the highest, since 
then the buckets are but partly filled and hi becomes small. Herein 
lies the great advantage of the overshot wheel; its disadvantage is in 
its large size and the expense of construction and maintenance. 






Breast Wheels. Art. 165 


437 


The number of buckets and their depth are governed by no laws 
except those of experience. Usually the number of buckets is about 
5 r or 6 r, if r is the radius of the wheel in feet, and their radial depth 
is from io to 15 inches. The breadth of the wheel parallel to its 
axis depends upon the quantity of water supplied, and should be so 
great that the buckets are not fully filled with water, in order that they 
may retain it as long as possible and thus make h x small. The wheel 
should be set with its outer circumference at the level of the tail water. 

Prob. 164 . Estimate the horse-power and efficiency of an overshot 
wheel which uses 1080 cubic feet of water per minute under a head of 26 
feet, the diameter of the wheel being 23 feet, and the water entering 15 0 
from the top and leaving 12 0 from the bottom. 

Art. 165. Breast Wheels 

The breast wheel is applicable to small falls, and the action of 
the water is partly by impulse and partly by weight. As repre¬ 
sented in Fig. 165 water 
from a reservoir is admit¬ 
ted through an orifice 
upon the wheel under the 
head h 0 with the velocity 
v 0 ; the water being then 
confined between the 
vanes and the curved 
breast acts by its weight 
through a distance /z 2 , 
which is approximately 
equal to h — //o, until 
finally it is released at the level of the tail race and departs with 
the velocity u, which is the same as that of the circumference of 
the wheel. The total energy of the water being Wh, the work 
of the wheel is eWh, if e be its efficiency. 

The reasoning of the last article may be applied to the breast 
wheel, hi being made equal to zero, and the expression there de¬ 
duced for e may be regarded as an approximate value of its the¬ 
oretic efficiency. It appears, then, that e will be the greater the 
smaller the fall ho ; but owing to leakage between the wheel and 
























438 


Chap. 13. Water Wheels 


the curved breast, which cannot be theoretically estimated, and 
which is less for high velocities than for low ones, it is not desir¬ 
able to make v 0 and h 0 small. The efficiency of the breast wheel is 
hence materially less than that of the overshot, and usually ranges 
from 50 to 80 percent, the lower values being for small wheels. 

Another method of determining the theoretic efficiency of the 
breast wheel is to discuss the action of the water in entering and 
leaving the vanes as a case of impulse. Let at the point of en¬ 
trance Av 0 and Au be drawn parallel and equal to the velocities 
Vo and u, the former being that of the entering water and the latter 
that of the vanes. Let a be the angle between v 0 and u, which 
may be called the angle of approach. Then the dynamic pressure 
exerted by the water in entering upon and leaving the vanes is, 
from Art. 158, = „ 7 g 0 cos« - u 

g 

and the work performed by it per second is 

_ JT/ (^0 coscc u)u 

g 

This expression has its maximum value when 

u — \v 0 cos« 

which gives the advantageous velocity of the wheel circumference, 
and the corresponding work of the dynamic pressure is 

k 0 = W v ° 2 cos * a 
4g 

Adding this to the work Wh 2 done by the weight of the water, 
the total work of the wheel when running at the advantageous 
velocity is found to be 

k = w M^a + h \ 

V 4 g / 

or, if i’o 2 be replaced by its value d 2 • 2 gh 0 , where c x is the coefficient 
of velocity for the stream as it leaves the orifice of the reservoir, 

k = W (ici 2 cos 2 « • ho + hi) 

whence the maximum hydraulic efficiency of the wheel is 

e = 4ci 2 cos 2 « • — + — 

h h 


(165) 






Undershot Wheels. Art. 166 


439 


If in this expression h 2 be replaced by h - h 0 , and if a = i and 
« = °°> this reduces to the same value as found for the overshot 
wheel. The angle a, however, cannot be zero, for then the direc¬ 
tion of the entering water would be tangential to the wheel, and 
it could not impinge upon the vanes; its value, however, should 
be small, say from io° to 25 0 . The coefficient c x is to be rendered 
large by making the orifice of the discharge with well-rounded 
inner corners so as to avoid contraction and the losses incident 
thereto. The above formulas cannot be relied upon in practice 
to give close values of k and e, on account of losses by foam and 
leakage along the curved breast, which of course cannot be al¬ 
gebraically expressed. 

Prob. 165 . A breast wheel is 10.5 feet in diameter, and has c\ = 0.93, 
// 0 = 4.2 feet, and a = 12 degrees. Compute the most advantageous num¬ 
ber of revolutions per minute. 


Art. 166 . Undershot Wheels 

The common undershot wheel has plane radial vanes, and the 
water passes beneath it in a direction nearly horizontal. It may 
then be regarded as a breast wheel where the action is entirely 
by impulse, so that in the preceding equations h 2 becomes o, 
lio becomes h , and a will be o°. The theoretic efficiency then is 
e = %c{ 2 . In the best constructions the coefficient C\ is nearly 
unity, so it may be concluded that the maximum efficiency of the 
undershot wheel is about 0.5. Experiments show that its actual 
efficiency varies from 0.20 to 0.40, and that the advantageous 
velocity is about 0.420 instead of 0.520. The lowest efficiencies 
are obtained from wheels placed in an unlimited flowing current, 
as upon a scow anchored in a stream ; and the highest from those 
where the stream beneath the wheel is confined by walls so as to 
prevent the water from spreading laterally. 

The Poncelet wheel, so called from its distinguished inventor, 
has curved vanes, which are so arranged that the water leaves 
them tangentially, with its absolute velocity less than that of 
the velocity of the wheel. If in Fig. 165 the fall h 2 be very small, 
and the vanes be curved more than represented, it will exhibit 


440 


Chap. 13. Water Wheels 


the main features of the Poncelet wheel. The water entering 
with the absolute velocity Vq takes the velocity u of the vane and 
the velocity V relative to the vane. Passing then under the wheel, 
its dynamic pressure performs work; and on leaving the vane 
its relative velocity V is probably nearly the same as that at 
entrance. Then if V be drawn tangent to the vane at the point 



of exit, and u tangent to the circumference, their resultant will 
be V\, the absolute velocity of exit, which will be much less than u. 
Consequently the energy carried away by the departing water 
is less than in the usual forms of breast and undershot wheels, 
and it is found by experiment that the efficiency may be as high 
as 60 percent. 

In Fig. 166 is shown a portion of a Poncelet wheel. At A 
the water enters the wheel through a nozzle-like opening with the 
absolute velocity vq and at B it leaves with the absolute velocity 
z>i. In the figure A and B have the same elevation. At A the 
entering stream makes the approach angle a with the circumfer¬ 
ence of the wheel and the same angle with the vane, so that the 
relative velocity V is equal to the velocity of the outer circum¬ 
ference u. If li be the head on A , the theoretic work of the water 
is W//, and the work of the wheel is 

k = w v ° 2 ~ Vl2 

2g 

and the efficiency, neglecting friction and leakage, is 

c _ Vo 2 ~Vi 2 
2 gh 

Now, let Ci be the coefficient of velocity of the entrance orifice, 











Vertical Impulse Wheels. Art. 167 


441 


then vq = Ci \ 2gh. From the parallelograms of velocity at 
A and B , there are found 

V 0 

u =-- V\ = 2 u since = vq tance 

2 cosa 

and for this velocity u the efficiency of the wheel is 

e = ci 2 (i — tan 2 «) ( 166 ) 

If Ci = i and a = o, the efficiency becomes unity. In the best 
constructions Ci may be made from 0.95 to 0.98, but a cannot 
be a very small angle, since then no water could enter the wheel. 
If « = 30° and Ci = 0.95 the efficiency is 0.60, which is probably 
a higher value than usually attained in practice. If the velocity 
be greater or less than %v 0 /cosa, the efficiency will be lowered on 
account of shock and foam at A. 

Prob. 166 . Estimate the horse-power that can be obtained from an 
undershot wheel with plane radial vanes placed in a stream having a mean 
velocity of 5 feet per second, the width of the wheel being 15 feet, its di¬ 
ameter 8 feet, and the maximum immersion of the vanes being 1.33 feet. 
How many revolutions per minute should this wheel make in order to furnish 
the maximum power ? 

Art. 167 . Vertical Impulse Wheels 

A vertical wheel like Fig. 166 , but having smaller vanes 
against which the water is delivered from a nozzle, is often called 
an impulse wheel, or a “ hurdy-gurdy ” 
wheel. The Pelton wheel, the Cascade 
wheel, and other forms can be purchased 
in several sizes and are convenient on ac¬ 
count of their portability. Figure 167 a 
shows an outline sketch of such a wheel 
with the vanes somewhat exaggerated in 
size. The simplest vanes are radial planes 
as at A, but these give a low efficiency. 

Curved vanes, as at B, are generally used, 
as these cause the water to turn back¬ 
ward, opposite to the direction of the motion, and thus to leave 
the wheel with a low absolute velocity (Art. 159 ). In the plan 
























442 


Chap. 13. Water Wheels 


of the wheel it is seen that the vanes may be arranged so as also 
to turn the water sidewise while deflecting it backward. The 
experiments of Browne * show that with plane radial vanes the 
highest efficiency was 40.2 percent, while with curved vanes or 
cups 82.5 percent was attained. The velocity of the vanes which 
gave the highest efficiency was in each case almost exactly one- 
half the velocity of the jet. 

The Pelton wheel is used under high heads, and also being of 
small size it has a high velocity. The effective head is that 
measured at the entrance of the nozzle by a pressure gage, cor¬ 
rected for velocity of 
approach and the loss in 
the nozzle by formula 
( 83 ) 1. These wheels are 
wholly of iron, and are 
provided with a casing 
to prevent the spattering 
of the water. Fig. 1676 
shows a form with three 
nozzles, by which three 
streams are applied at 
different parts of the 
circumference, in order 
to obtain a greater power 
than by a single nozzle, 
or to obtain a greater 
speed by using smaller 
nozzles. For an effective head of 100 feet and a single nozzle 
the following quantities are given by the manufacturers : 



Diameter in feet, 

1 

2 

3 

4 

6 

Cubic feet per minute, 

8.29 

44.19 

99-52 

176.7 

398.1 

Revolutions per minute 

;, 726 

363 

242 

181 

121 

Horse-powers, 

1.40 

7-49 

16.84 

29-93 

67-3 

and these figures imply an 

efficiencv of 

j 

85 percent. 



The general theory of these vertical impulse wheels is the same 
as that given for moving vanes in Art. 158 . Owing to the high 


* Bowie’s Treatise on Hydraulic Mining (New York, 18S5), p. 193. 























































Horizontal Impulse Wheels. Art. 168 


443 


velocity, more or less shock occurs at entrance, and as the angle 
of exit /3 cannot be made small, the water leaves the vanes with 
more or less absolute velocity. The advantageous velocity of 
the vanes or cups is between 40 and 50 percent of that of the 
entering jet. 

Prob. 167. The diameter of a hurdy-gurdy wheel is 12.5 feet between 
centers of vanes, and the impinging jet has a velocity of 58.5 feet per second 
and a diameter of 0.182 feet. The efficiency of the wheel is 44.5 percent, 
when making 62 revolutions per minute What effective horse-power does 
it furnish? 


Art. 168 . Horizontal Impulse Wheels 
When a wheel is placed with its plane horizontal and is driven 


by a stream of water from a nozzle, it is called a horizontal im¬ 
pulse wheel. There are two forms, known as the outward-flow 
and the inward-flow wheel. In the former, shown in Fig. 168 a, 
the water enters the wheel upon the inner and leaves it upon the 



Fig. 168 b. 


Fig. 168a. 


outer circumference; in the latter, shown in Fig. 168 b, the water 
enters upon the outer and leaves upon the inner circumference. 
The water issuing from the nozzle with the velocity v impinges 
upon the vanes, and in passing through the wheel alters both its 
direction and its absolute velocity, thus transforming its energy 
into useful work. The energy of the entering water is W • 2g 

and that of the departing water is W • V\ 2 /2g. Neglecting fric¬ 
tional resistances, the work imparted to the wheel by the water is 







444 


Chap. 13. Water Wheels 


and dividing this by the theoretic energy, the efficiency is 

e—i — OiA ) 2 

This is the same as the general formula ( 163 ) if h' = o; that is, if 
losses in foam and friction are disregarded, and if the wheel is 
set at the level of the tail race. It is now required to state the 
conditions which will render these losses and also the velocity 
V\ as small as possible. The reasoning will be general and appli¬ 
cable to both outward and inward-flow wheels. 

At the point A where the water enters the wheel let the paral¬ 
lelogram of velocities be drawn, the absolute velocity of entrance 
being resolved into its two components, the velocity u of the wheel 
at that point, and the velocity V relative to the vane; let the 
approach angle between u and v be called «, and the entrance 
angle between u and V be called </>. At the point B where the 
water leaves the wheel let \\ be its velocity relative to the vane, 
and ii\ the velocity of the wheel at that point; then their result¬ 
ant is »i, the absolute velocity of exit. Let the exit angle between 
Vi and the reverse direction of n\ be called ft. The directions of 
the velocities u and U\ are of course tangential to the circumfer¬ 
ences at the points A and B. Let r and r i be the radii of these 
circumferences; then the velocities of revolution are directly as 
the radii, or uy\ = U\Y. 

In order that the water may enter the wheel without shock 
and foam, the relative velocity V should be tangent to the vane 
at A, so that the water may smoothly glide along it. This will 
be the case if the wheel is run at such speed that the parallelo¬ 
gram at A can be formed, or when the velocities u and v are pro¬ 
portional to the sines of the angles opposite them in the triangle 
Auv. The velocity Vi will be rendered very small by running the 
impulse wheel at such speed that the velocities u\ and Vi are 
equal, since then the parallelogram at B becomes a rhombus and 
the diagonal V\ is very small. Hence 

u sin (<b — a) . T _ 

- = —— and Ui = V i 

v sin 9 

are the two conditions of maximum hydraulic efficiency. 


(168)! 




Horizontal Impulse Wheels. Art. 168 445 

Now, referring to the formula ( 160 ), which expresses the re¬ 
lation between the velocities of rotation and the relative velocities 
of the water for revolving vanes, it is seen that if u\ = V x , then 
also u = V. But u cannot equal V unless </> = 2a, and then 
u = v/ 2 cos a, which is the advantageous velocity of the circum¬ 
ference at A. Therefore the two conditions above reduce to 


</> = 2a and u = —-— ( 168) 2 

2 COStt 

which show how the wheel should be built and what speed it 
should have to secure the greatest efficiency. When this speed 
obtains, the absolute velocity v\ is 

• l/Q n • i Q r\ sin h3 
Vi = 2Ui sin f ft = 2 u— sin %ft = v— -— 

r r cos« 


and the corresponding hydraulic efficiency is 

(r\ sinJ^V 


i — 


V 


r cos a 


( 168 ), 


by the discussion of which proper values of the approach angle a 
and the exit angle /3 can be derived. 

This formula shows that both the approach angle a and the 
exit angle /3 should be small in order to give high efficiency, but 
they cannot be zero, as then no water could pass through the 
wheel; values of from 15 0 to 30° are usual in practice. It 
also shows that /3 is more important than «, and if ft be small, « 
may sometimes be made 40° or 45 0 . It likewise shows that for 
given values of a and ft the inward-flow wheel, in which ri is less 
than r, has a higher efficiency than the outward-flow wheel. 

The condition V l = u x renders the absolute exit velocity i\ very 
small, but it does not give its true minimum. This will be obtained 
by making V x = u x cos ft, so that the direction of v x is normal to that 
of V h and thus v x — u x sin ft. The discussion of water wheels and tur¬ 
bines under this condition of the true minimum leads to very complex 
formulas, and hence in this book, as in many others, the simpler con¬ 
dition V x = u x is used. 

Prob. 168. Compute the maximum efficiency of an outward-flow im¬ 
pulse wheel when ri = 3 feet, r = 2 feet, a = 45 0 , <^> = 90°, (3 = 30°, and 





446 


Chap. 13. Water Wheels 


find the number of revolutions per minute required to secure such effi¬ 
ciency when the velocity of the entering stream is v = ioo feet per second. 

Art. 169. Downward-flow Impulse Wheels 

In the impulse wheels thus far considered the water leaves the 
vanes in a horizontal direction. Another form used less frequently 
is that of a horizontal wheel driven by water issuing from an in¬ 
clined nozzle so that it 
passes downward along the 
vanes without approaching 
or receding from the axis. 
Figure 169 shows an out¬ 
line plan of such an 
impulse wheel and a de¬ 
velopment of a part of a 
cylindrical section. Let v 
be the velocity of the en¬ 
tering stream, u that of the 
wheel at the point where it 
strikes the vanes, and vi 
the absolute velocity of the 
departing water. At the 
entrance A the direction 
of v makes with that of u 
the approach angle a , and 
the direction of the rela¬ 
tive velocity V makes with 
that of u the entrance angle <f>. The water then passes over 
the vane, and, neglecting the influence of friction and gravity, it 
issues at B with the same relative velocity V, making the exit 
angle /3 with the plane of motion. 

The condition that impact and foam shall be avoided at A 
is fulfilled by making the relative velocity V tangent to the vane, 
and the condition that the absolute velocity v x shall be small is 
fulfilled by making the velocities u and V equal at B. Hence, 
as in the last article, the best construction is to make <£ = 2 «, 














Downward-flow Impulse Wheels. Art. 169 


447 


and the best speed of the wheel is u = v/ 2cos«. Also by the 

same reasoning the efficiency under these conditions is 


e —i — (sin J/ 3 /cos«) 2 


which shows that a , and especially / 3 , should be a small angle to 
give a high numerical value of e. For instance, if both these 
angles are 30°, the efficiency is 0.92, but if a = 45 0 and /3 = io°, 
the efficiency is 0.94. 

Although these wheels are but little used, there seems to be no 
hydraulic reason why they should not be employed with a success 
equal to or greater than that attained by vertical impulse wheels. 
It will be possible to arrange several nozzles around the circumference 
and thus to secure a high power with a small wheel. The fall of the 
water through the vertical distance between A and B will also add 
slightly to the power of the wheel, and if this be taken into account, 
the above values of advantageous velocity and efficiency will be modi¬ 
fied, both being slightly increased, as the following investigation shows. 

Let hi be the vertical fall between A and B ; then the theoretic 
energy of the water with respect to B is 



and the hydraulic efficiency of the wheel is 

e _ 1 _ fli 2 

v 2 + 2ghi 

Here the relative velocity V 1 at B is greater than V, or 

Vi 2 = V 2 + 2 gh 

and since u should equal V h this equation becomes, after inserting 
for V its value in terms of u, v, and «, 



which gives the advantageous velocity of the wheel. Since 

Vi = 2u sin J/S, 

the above expression for the theoretic hydraulic efficiency reduces to 







448 


Chap. 13. Water Wheels 


For this case the approach angle <f> must be a little greater than 2a, 
and its value can be found by 

cot <f> = cot« — ' 2 ^ 1 - 

v 2 sin2a 

and by using this angle <£, losses due to impact will be avoided when 
the wheel is run at the advantageous speed. For example, if v = 50 
feet per second, and h l = 1 foot, and a = 30°, the value of <f> is about 
63° instead of 6o° as the simpler condition requires, while the increase 
in the advantageous speed is about 2 percent over the former value. 

Prob. 169 . A wheel like Fig. 169 is driven by water which issues from 
a nozzle with a velocity of 100 feet per second. If the diameter is 3 feet, the 
efficiency 0.90, and the approach angle a = 45 0 , find the best value of the 
entrance and exit angles and the best speed. 


Art. 170. Nozzles for Impulse Wheels 


Impulse wheels are driven by the dynamic pressure of water 
issuing from nozzles attached to the end of a pipe which conducts 
the water from a reservoir. It is shown in Art. 101 that the 
greatest velocity is secured when the diameter of the nozzle is as 
small as possible and that the greatest discharge occurs when there 
is no nozzle. To secure the greatest power, however, there is a 
certain diameter of nozzle which will now be determined, and it is 
advisable for economical reasons to use a nozzle of this size and 
adjust the speed of the wheel thereto. 

Let h be the hydrostatic head on the nozzle, l the length, and 
d the diameter of the pipe, and D the diameter of the nozzle. 
Let all the resistances except that due to friction in the pipe and 
nozzle be neglected; then from Art. 101 the velocity of the jet 
from the nozzle is 


V = 


ish 


i/(i/d)(D/dy+( i/ Cl y 

in which / is the friction factor for the pipe and C\ is the coefficient 
of velocity for the nozzle. Let w be the weight of a cubic foot 
of water; then the theoretic energy of the jet per second is 


4 §_ 

“ 2 


K = w ■ \ ttDW ■ P- = ™ (ISKhlDl) 

2 g 8 g\fc 1 HD i + d ! ’J 








Nozzles for Impulse Wheels. Art. 170 


449 


and the value of D which renders this a maximum is, by the usual 
method of differentiation, ascertained to be 

D = {<P/2fc 1 H) k ( 170 )! 

and for a nozzle of this size the velocity of the jet is 

V = o.816 Cl's/ 2 gh 

or, since C\ is about 0.97, the velocity of the jet when leaving the 
nozzle is about 80 percent of the theoretic velocity due to the 
head on the nozzle. 

As an example let a pipe be 1200 feet long and 1 foot in 
diameter; then, taking for / the mean value 0.02 and using 
Ci = 0.97, there is found D = 0.39 feet, and hence a nozzle 4! 
inches in diameter is required to give the maximum power. This 
result may be revised, if thought necessary, by finding the velocity 
in the pipe and thus getting a better value of / from Table 90 a. 
If the head be 100 feet, this velocity is found to be 9.2 feet per 
second, whence / = 0.018, and on repeating the computation 
there is found D = 0.40 feet = 4.8 inches. If the pipe be 12 000 
feet long, the advantageous diameter of the nozzle will be found 
to be much smaller, namely, 2\ inches. 

When there is more than one nozzle at the end of the pipe, the above 
investigation must be modified. Let there be two nozzles with the 
diameters and D 2 , each having the coefficient c x . Then the dis¬ 
charge rd 2 v through the pipe equals the discharge \^{D^Vi + Z) 2 2 F 2 ). 
But the velocities Vj and V 2 are equal if the tips of the nozzles are on 
the same elevation, and hence d 2 v equals (Lff 2 -f D 2 2 ) V, where V is 
the velocity of flow from each nozzle. Now, referring to Art. 101 
and to the proof of ( 170 ), it is seen that it applies to this case provided 
D 2 be replaced by Zfi 2 + D 2 2 , and accordingly 

Di 2 + ZV = 0 d 5 / 2 fci 2 l )* ( 170) 2 

is the formula for determining the sizes of the two nozzles which will 
furnishr the maximum power; if D x be assumed, the value of D 2 can 
be computed. The area of the circle of diameter D found from 
( 170 )i is equal to the sum of the areas of the two circles found from 
( 170 ) 2 . If there be three or more nozzles, the sum of their areas is 
equal to that corresponding to the diameter D as computed from 



450 


Chap. 13. Water Wheels 


( 170 X. For example, let there be a pipe 1200 feet long and one foot 
in diameter to which three nozzles of equal size are attached. The 
diameter found above for one nozzle is 4.80 inches, and the correspond¬ 
ing area is 18.10 square inches; hence the area of the cross-section 
of the tip of each of the three nozzles is 6.03 square inches, which cor¬ 
responds to a diameter of 2.77 inches. 

Prob. 170 . A pipe 15 000 feet long and 18 inches in diameter runs from 
a mountain reservoir to a power plant, where the water is to be delivered 
through two nozzles against a hurdy-gurdy wheel. If the diameter of one 
nozzle is 2 inches, find the diameter of the other in order that the maximum 
power may be developed. If the head on the nozzles is 623 feet and the 
efficiency of the wheel 79 percent, compute the horse-power that may be 
expected. 

Art. 171 . Special Forms of Wheels 

Numerous varieties of the water wheels above described have 
been used, but the variation lies in mechanical details rather 
than in the introduction of any new hydraulic principles. In 
order that a wheel may be a success it must furnish power as 
cheap as or cheaper than steam or other motors, and to this 
end compactness, durability, and low cost of installation and 
maintenance are essential. 

A variety of the overshot wheel, called the back-pitch wheel, 
has been built, in which the water is introduced on the back instead 
of on the front of the wheel. The buckets are hence differently 
arranged from those of the usual form, and the wheel revolves 
also in an opposite direction. One of the largest overshot wheels 
ever constructed is at Laxey, on the east coast of the Isle of 
Man. It is 72J feet in diameter, about 10 feet in width, and 
furnishes about 150 horse-power, which is used for pumping 
water out of a mine. 

A breast wheel with very long curved vanes extending over 
nearly a fourth of the circumference has been used for small falls, 
the water entering directly from the penstock without impulse, 
so that the action is that of weight alone. This form is made of 
iron and gives a high efficiency. 

Undershot wheels with curved floats for use in the open cur¬ 
rent of a river have been employed, but in order to obtain much 


Special Forms of Wheels. Art. 171 


451 


power they require to be large in size, and hence have not been 
able to compete with other forms. The great amount of power 
wasted in all rivers should, however, incite inventors to devise 
wheels that can economically utilize it. Currents due to the 
movement of the tides also afford opportunity for the exercise 
of inventive talent. 

The conical wheel, or danaide, is an ancient form of down¬ 
ward-flow impulse wheel, in which the water approaches the axis 
as it descends, and thus its relative motion is decreased by the 
centrifugal force. The theory of this is almost precisely the same 
as that of an inward-flow impulse wheel, and there seems to be 
no hydraulic reason why it should not give a high efficiency. 
Another form of danaide has two or more vertical vanes attached 
to an axis, which are inclosed in a conical case to prevent the 
lateral escape of the water. 

A water-pressure engine is a hydraulic motor which moves under 
the static pressure of water acting against a piston or a revolving 
disk. The piston forms are reciprocating in motion like the steam- 
engine and operate in the same way, the water entering and leaving 
through ports which are opened and closed by a link motion con¬ 
nected with the piston-rod. The other forms give rotary motion 
directly from the revolving vanes or disks. The piston engine has 
been employed in Germany to a considerable extent to drive pumps 
for draining mines, but the rotary engine has not been widely used, 
and it cannot be advantageously arranged to deliver a high power. 
On account of the incompressibility of water, special devices for 
regulating the opening and closing of the valves are necessary. 

Numerous other special devices for utilizing the energy of water 
by means of water wheels have been invented, but they do not in¬ 
troduce any new hydraulic principle. The efficiency of these special 
forms is often low on account of the imperfections of the apparatus, 
but it should be borne in mind that high efficiency is only obtained after 
trials extending over much time, such trials enabling the imper¬ 
fections to be discovered and removed. The formulas for hydraulic 
efficiency deduced in the preceding pages do not include losses due to 
friction, and these may often amount to io or 20 percent of the 
theoretic energy, so that due allowance for them should be made in 
estimating the power which a proposed design may deliver. 


452 


Chap. 13. Water Wheels 


Power may be obtained from the ocean waves, which are constantly 
rising and falling, by a suitable arrangement of wheels and levers, and 
some inventions in this direction have given fair promise of success. 
One in operation on the coast of England about 1890 consisted of a 
large buoy which rose and fell with the waves on a fixed vertical 
shaft fastened in the rock bottom. As the buoy moved up and down 
it operated a system of levers and wheels which drove an air-compressor, 
and this in turn ran a dynamo that generated electric power. The rise 
of the ocean tide also affords opportunity for impounding water which 
may be used to generate power when the tide falls. 'Plants for this 
purpose are to be located along tidal rivers where opportunities for 
impounding occur, the wheels being idle during the rise of the tide, 
and in operation during its fall. Owing to this intermittent gener¬ 
ation of power, it will be necessary to provide for its storage, so that 
industries using it may be in continuous operation. 

Prob. 171 a. A wheel using 10.5 cubic meters of water per minute under 
an effective head of 23.4 meters has an efficiency of 75 percent. What metric 
horse-power does it deliver ? What is its power in kilowatts ? 

Prob. 1716 . A breast wheel has = 0.95, // 0 =1.3 meters, and a = 12 0 . 
If its diameter is 3.5 meters, compute the most advantageous number of 
revolutions per minute. 

Prob. 171 c. An inward-flow impulse wheel has <P = 104°, a = 52 0 , 
and (3 = 12 0 , its inner diameter being 0.82 meters and its outer diameter 
1.22 meters. If this wheel uses 0.86 cubic meters of water per second under 
an effective head of 7.9 meters, compute its efficiency and its probable effec¬ 
tive horse-power. 

Prob. \~Ild. A pipe 3200 meters long and 40 centimeters in diameter 
delivers water through two nozzles against a hurdy-gurdy wheel. When the 
diameter of one nozzle is 5 centimeters, find the diameter of the other nozzle 
in order that the energy of the two jets may be a maximum. If the head 
on the nozzles is 107 meters and the efficiency of the wheels is 81 percent, 
compute the horse-power which the wheels will deliver. 


The Reaction Wheel. Art. 172 


453 


• CHAPTER 14 

\ . , 

TURBINES 

Art. 172 . The Reaction Wheel 

The reaction wheel, invented by Barker about 1740, consists 
of a number of hollow arms connected with a hollow vertical shaft, 
as shown in Fig. 172 . The water issues from the ends of the 
arms in a direction opposite to that of their motion, and by 
the dynamic pressure due to its reaction 

the energy of the water is transformed x# j / 

into useful work. Let the head of water 
CC in the shaft be h ; then the pressure- 
head BB which causes the flow from 
the arms is greater than h , on account 
of the centrifugal force due to the rota¬ 
tion of the wheel. Let U\ be the abso¬ 
lute velocity of the exit orifices, and V\ 
be the velocity of discharge relative to 
the wheel; then, as shown in Art. 29 , 
and also in Art. 162 , 

Vi = V 2gh + Ui 2 

The absolute velocity 24 of the issuing 
water now is 

Vi = V\ — Ui = \ 2gh J \- U\ — u\ 

It is seen at once that the efficiency can never reach unity unless 
V\ = o, which requires that V\ = U\. This, however, can only 
occur when U\ = 00 , since the above formula shows that V\ must 
be greater than U\ for any finite values of h and U\. To de¬ 
duce an expression for the efficiency the work of the wheel 



Fig. 172. 
















454 


Chap. 14. Turbines 


WQi — vi 2 /2g) is to be divided by the theoretic energy of the 
water Wh, and this gives 


V\ _ (V l~Ui) 2 _ 2 Ui 

2 gh V i — U\ 2 V i + iii 


( 172 ): 


which shows, as before, that e equals unity when IT = U\ = oo. 
If Vi = 2U\, the value of e is 0.667 5 if Pi = 3^1, the value of e 
is reduced to 0.50. 

This investigation indicates that the efficiency of a reaction 
wheel increases with its speed. If a x be the area of the exit orifices 
and w the weight of a cubic unit of water, the weight of the water 
discharged in one second is wa\V 1, which becomes infinite when 
V\ = U\ = 00 . Nothing approaching this can be realized, and 
on account of losses due to friction, a very high speed is imprac¬ 
ticable. The reaction wheel, indeed, is like the jet propeller in 
regard to efficiency (Art. 186 ). 


To consider the effect of friction in the arms, let c x be the coefficient 
of velocity (Chap. 7 ), so that 

V1 = Ci~V 2 ghui 1 

Then the effective work of the wheel is 

k = W fa ^ + ^i 2 ~ u i) ui 

g 

and the corresponding efficiency of the wheel is 

_ C1U1 V2 gh + u \ 2 — iii 2 
gh 

The value of u u which renders this a maximum, is 

ui~ = ~ gh 

V1 - ci 2 

and this reduces the value of the efficiency to 

e= 1-Vi-ci 2 ( 172 ), 

If c\ = 1, there is no loss in friction, and u x = co and e — 1, as be¬ 
fore deduced. If c x = 0.94, the advantageous velocity u x is very nearly 
V2 gh, and e is 0.66; hence the influence of friction in diminishing the 
efficiency is very great. In order to make c x large, the end of the arm 














Classification of Turbines. Art. 173 


455 


where the water enters must be well rounded to prevent contraction, 
and the interior surface must be smooth. If the inner end has sharp, 
square edges, as in a standard tube (Art. 78 ), c x is 0.82, and e is 0.43. 

The reaction wheel is not now used as a hydraulic motor on account 
of its low efficiency. Even when run at high speeds the efficiency 
is low on account of the greater friction and resistance of the air. 
By experiments on a wheel one meter in diameter under a head of 
1.3 feet Weisbach found a maximum efficiency of 67 percent when the 
velocity of revolution u x was V2 gh. When u x was 2V2 gh, the efficiency 
was nothing, or all the energy was consumed in frictional resistances. 

The reaction wheel is here introduced at the beginning of the dis¬ 
cussion of turbines mainly to call attention to the fact that the dis¬ 
charge varies with the speed. Although sometimes called a turbine, 
it can scarcely be properly considered as belonging to that class of 
hydraulic motors. 

Prob. 172 . The sum of the exit orifices of a reaction wheel is 4.25 
square inches, their radius is 1.75 feet, and their velocity 32.1 feet, per 
second. Compute the head necessary to furnish 1.6 horse-powers, when 
ci = 0.95. 

Art. 173 . Classification of Turbines 

A turbine wheel may be defined as one in which the water 
enters around the entire circumference instead of upon one por¬ 
tion, so that all the moving vanes are simultaneously acted upon 
by the dynamic pressure of the water as it changes its direction 
and velocity. The turbine was invented by Fourneyron in 1827, 
and owing to its compactness, cheapness, and high efficiency, it 
has largely replaced the older forms of water wheels. Turbines 
are usually horizontal wheels, and like the impulse wheels of the 
last chapter, they may be outward-flow, inward-flow, or down¬ 
ward-flow, with respect to the manner in which the water passes 
through them. In the outward-flow type the water enters the 
wheel around the entire inner circumference and passes out around 
the entire outer circumference (Fig. 1745 ). In the inward-flow 
type the motion is the reverse (Fig. 174 c). In the downward- 
flow type the water enters around the entire upper annular 
openings, passes downward between the moving vanes, and 
leaves through the lower annulus (Fig. 179 a). In all cases the 




456 


Chap. 14. Turbines 


water in leaving the wheel should have a low absolute velocity, so 
that most of its energy may be surrendered to the turbine in the 
form of useful work. 

The supply of water to a turbine is regulated by a gate or 
gates, which can partially or entirely close the orifices where the 
water enters or leaves. The guides and wheel, with the gates 
and the surrounding casings, are made of iron. Numerous forms 
with different kinds of gates and different proportions of guides 
and vanes are in the market. They are made of all sizes from 
6 to 6o inches in diameter, and larger sizes are built for special 
cases. The great turbines at Niagara are of the outward-flow 
type, the inner diameter of a wheel being 63 inches and each twin 
turbine furnishing about 5000 horse-powers (Art. 182 ). The 
smaller sizes of turbines used in the United States are mostly of 
the inward-flow type or of a combined inward- and downward- 
flow type. 

The three typical classes of turbines above described are often 
called by the names of those who first invented or perfected them ; 
thus the outward-flow is called the Fourneyron, the inward-flow 
the Francis, and the downward-flow the Jonval turbine. There 
are also many turbines in the market in which the flow is a com¬ 
bination of inward and downward motion, the water entering 
horizontally and inward, and leaving vertically, the vanes being 
warped surfaces. The usual efficiency of turbines at full gate 
is from 70 to 85 percent, although 90 percent has in some cases 
been derived. When the gate is partly closed, the efficiency in 
general decreases, and when the gate opening is small, it becomes 
very low. This is due to the loss of head consequent upon the 
sudden change of cross-section; and therein lies the disadvan¬ 
tage of the turbine, for when the water supply is low, it is im¬ 
portant that it should utilize all the power available. A com¬ 
pilation of turbine tests with descriptions of the various forms 
of wheels has been made by Horton and issued by the United 
States Geological Survey.* 

Another classification is into impulse and reaction turbines. 

* Water Supply and Irrigation Paper, No. 180, 1906. 


Reaction Turbines. Art. 174 


457 


In an impulse turbine the water enters the wheel with a velocity 
due to the head at the point of entrance, just as it does from the 
nozzle which drives an impulse wheel (Art. 168 ). In a reaction 
turbine, however, the velocity of the entering water may be 
greater or less than that due to the head on the orifices of entrance, 
and, as in the reaction wheel, it is also influenced by the speed. 
This is due to the fact that in a reaction turbine the static pres¬ 
sure of the water is partially transmitted into the moving wheel, 
provided that the spaces between the vanes are fully filled. Any 
turbine may be made to act either as an impulse or a reaction 
turbine. If it be arranged so that the water passes through the 
vanes without filling them, it is an impulse turbine; if it be 
placed under water, or if by other means the flowing water is 
compelled to completely fill all the passages, it acts as a reaction 
turbine. As will be seen later, the theory of the reaction turbine 
is quite different from that of the impulse turbine. 

Prob. 173 . If the efficiency of a turbine is 75 percent when delivering 
5000 horse-powers under a head of 136 feet, how many cubic feet of water per 
minute pass through it ? 

/ 

Art. 174 . Reaction Turbines 

A reaction turbine is driven by the dynamic pressure of 
flowing water which at the same time may be under a certain 
degree of static pres¬ 
sure. If in the reaction 
wheel of Fig. 172 the 
arms be separated from 
the penstock at A, and 
be so arranged that BA 
revolves around the axis 
while AC is stationary, 
the resulting apparatus 
may be called a reac¬ 
tion turbine. The static 
pressure of the head CC 
can still be transmitted 

through the arms, so Fig. 174a. 




































































458 


Chap. 14. Turbines 


that, as in the reaction wheel, the discharge will be influenced 
by the speed of rotation. The general arrangement of the 
moving part is, however, like that of an impulse wheel, the 
vanes being set between two annular frames, which are attached 



by arms to a central axis. In Fig. 174 a is a vertical section 
showing an outward-flow wheel W to which the water is 
brought by guides G from a fixed penstock P. Between the 
guides and the wheel there is an annular space in which slides 



Fig. \ 7 U . 


































































































































































































































































Reaction Turbines. Art. 174 


459 


an annular vertical gate E ; this serves to regulate the quantity 
of water, and when it is entirely depressed, the wheel stops. 
Many other forms of gates are, however, used in the different 
styles of turbines found in the market. 



In Figs. 1745 and 174c are given horizontal and vertical sec¬ 
tions of both the outward- and the inward-flow types, showing 
the arrangement of guides and vanes. The fixed guide passages 
which lead the water from the penstock are marked G, while the 
moving wheel is marked W. It is seen that the water is intro¬ 
duced around the entire circumference of the wheel, and hence 
the quantity supplied, and likewise the power, is far greater than 
in the impulse wheels of the last chapter. 

In order that the static pressure may be transmitted into the 
wheel it is placed under water, as in Fig. 174 a, or the exit orifices 
are partially closed by gates, or 
the air is prevented from enter¬ 
ing them by some other device. 

In Fig. 174 d a Leffel turbine 
of the inward-flow type is illus¬ 
trated, the arrows showing the 
direction of the water as it enters 
and leaves. The wheel itself is 
not visible, it being within the 
inclosing case through which the 
water enters by the spaces be¬ 
tween the guides. In Fig. 174c 
is shown a view of a Hunt tur¬ 
bine, which is also of the inward- 
and downward-flow type. In 
both cases the guides are seen 
with the small shaft for moving 
the gates, these being partly raised 
in Fig. 174c. The flange at the 
base of the guides serves to sup¬ 
port the weight of the entire apparatus upon the floor of the 
inclosing penstock, which is filled with watei to the level of 











































































4G0 


Chap. 14 . Turbines 


the head bay. The cylinder below the flange, commonly called 
a draft-tube, carries away the water from the wheel, and the 
level of the tail water should stand a little higher than its 
lower rim in order to prevent the entrance of air and thus in¬ 
sure that the wheel may act as a reaction turbine. Iron pen¬ 
stocks are frequently used instead of wooden ones, and for the 
pure outward- and inward-flow types the wheel is often placed 
below the level of the tail race. 

Turbines are sometimes placed vertically on a horizontal 
shaft. Fig. 174/ shows twin Eureka turbines thus arranged in 



an inclosing iron casing. The water enters through a large 
pipe attached to the cylinder opening, and having filled the 
cylindrical casing, it passes through the guides, turns the wheels, 
and escapes by the two elbows. Large twin vertical turbines fur¬ 
nishing 1200 horse-powers have been installed at Niagara Falls 
by the James Leffel Company. 

All reaction turbines will act as impulse turbines when from 
any cause the passages between the vanes, or buckets, as they 
are generally called, are not filled with water. In this case the 
theory of their action is exactly like that of the impulse wheels 
described in the last chapter. In Arts. 175-178 reaction turbines 
of the simple outward- and inward-flow types will be discussed, 
the downward-flow type being reserved for special description 
in Art. 179. 

Prob. 174 . Consult Engineering Record, Feb. 5, 1898, and describe 
methods of regulating the speed of turbines. 















































Flow through Reaction Turbines. Art. 175 


461 


Art. 175. Flow through Reaction Turbines 


The discharge through an impulse turbine, like that for an 
impulse wheel, depends only on the area of the guide orifices and 
the effective head upon them, or q =av = a^/ 2 gh. In a re¬ 
action turbine, however, the discharge is influenced by the speed 
of revolution, as in the reaction wheel, and also by the areas of 
the entrance and exit orifices. To find 
an expression for this discharge let the 
wheel be supposed to be placed below 
the surface of the tail water, as in Fig. 

175. Let h be the total head between 
the upper water level and that in the 
tail race, Hi the pressure-head on the exit 
orifices, and H the pressure-head at the 
gate opening as indicated by a piezom¬ 
eter supposed to be there inserted. Let 
U\ and u be the velocities of the wheel at 
the exit and entrance circumference, which have radii Y\ and r 
(Fig. 1746). Let V\ and V be the relative velocities of exit and 
entrance, and v 0 be the absolute velocity of the water as it 
leaves the guides and enters the wheel; the entering velocity Vq 
may be less or greater than V 2 gh, depending upon the value 
of the pressure-head H . Let a\, a, and a 0 be the areas of the 
orifices normal to the directions of V\, V, and v 0 . Now, neglect¬ 
ing all losses of friction between the guides, the theorem of 
Art. 31, that pressure-head plus velocity-head equals the total 
head, gives the equation 

H + '^ = h + H l 
2g 



Fig. 175. 


Also, neglecting the friction and foam in the buckets, the corre¬ 
sponding theorem of Art. 162 gives 

ffl+ JV-ifii.fl+E-*! 

2 g 2 g 2 g 2 g 

Adding these equations, the pressure-heads Hi and H disappear, 
and there results the formula 

Vi t —V 2 +Vo 2 = 2 gh + Ui — u 2 ( 175 )! 








































462 


Chap. 14 . Turbines 


Now, since the buckets are fully filled, the same quantity of 
water, q, passes in each second through each of the areas ai, 
a, and a 0 , and hence the three velocities through these areas have 
the respective values, 

Vi = ±, F = 2 , 00 = 1 

a i a a 0 


Introducing these values into the formula (175)i, solving for q, 
and multiplying by a coefficient c to account for losses in leakage 
and friction, the discharge per second is 



(175), 


This is the formula for the flow through a reaction turbine when 
the gate is fully raised. The reasoning applies to an inward-flow 
as well as to an outward-flow wheel. In an outward-flow turbine 
U\ is greater than u , and consequently the discharge increases 
with the speed; in an inward-flow turbine U\ is less than u, and 
consequently the discharge decreases as the speed increases. 

The value of the coefficient c will usually vary with the head, and 
also with the size of the areas a h a, and a 0 . When a turbine has been 
tested by the methods of Arts. 147-150, and the areas have been meas¬ 
ured, the values of c for different speeds may be computed. For 
example, take the outward-flow Boyden turbine, tests of which at full 
gate are given in Art. 150. The measured dimensions and angles of 
this wheel are as follows : 


Outer radius of wheel i\ = 

Inner radius of wheel r = 

Outer radius of guide case r 0 = 

Outer depth of buckets d\ = 

Inner depth of buckets cL = 

Outer area of buckets a { = 

Inner area of buckets a = 

Outer area of guide orifices a 0 = 

Exit angle of buckets (3 = 

Entrance angle of buckets cf> = 

Entrance angle of guides a = 


3.3167 feet 
2.6630 feet 
2.5911 feet 
0.722 feet 
0.741 feet 
4.61 square feet 
12.12 square feet 
4.76 square feet 
13.5 degrees 
90 degrees 
24 degrees 


Number of buckets 52 , number of guides, 32 





Flow through Reaction Turbines. Art. 175 


463 


Inserting in the above formula the values of a h a, and a 0 , placing for 
n 2 —u 2 its value (^oWV) 2 (r x 2 —r 2 ), where N is the number of revolutions 
per minute, it reduces to 

q = 3.44cV2g/z-f- 0.04 8A 2 

From this the value of c may be computed for each of the seven exper¬ 
iments, and the following tabulation shows the results, the first four 
columns giving the number of the experiment, the observed head, num¬ 
ber of revolutions per minute, and discharge in cubic feet per second. 
The fifth column gives the theoretic discharge computed from the 
above formula, taking the coefficient as unity, and the last column 
is derived by dividing the observed discharge q by the theoretic dis¬ 
charge Q. The discrepancy of 5 or 6 percent is smaller than might 
be expected, since the formula does not consider frictional resistances. 


No. 

h 

N 


Q 

c 

21 

17.16 

63-5 

117.01 

123-1 

0.950 

20 

17.27 

70.0 

118.37 

125.2 

0.945 

19 

17-33 

75-o 

H9-53 

126.8 

0.943 

18 

17-34 

80.0 

121.15 

128.4 

0.944 

17 

17.21 

86.0 

122.41 

130.0 

0.942 

16 

17.21 

93-2 

124.74 

132-5 

0.941 

15 

17.19 

100.0 

127.73 

134-9 

0.947 


A satisfactory formula for the discharge through a turbine when 
the gate is partly depressed is difficult to deduce, because the loss of 
head which then results can only be expressed by the help of experi¬ 
mental coefficients similar to those given in Art. 92 for the sliding gate 
in a water pipe, and the values of these for turbines are not known. 
It is, however, certain that for each particular gate opening the dis¬ 
charge is given by /— 7—7 « 2 

in which m depends upon the areas of the orifices and the height to 
which the gate is raised. For instance, in the tests of the above Boy- 
den turbine the mean value of m for full gate opening is 3.25, but when 
the gate was only six-tenths open, its value was 2.81, and when the 
gate was two-tenths open, its value was 1.36. Each form and size ol 
reaction turbine has its own values of ni, depending upon the areaol its 
orifices, and w r hen these have been determined, a turbine may be used 
as a water meter to measure the discharge with a fair degree of precision. 

Prob. 175 . Consult Francis’ Lowell Hydraulic Experiments, pp. 67-75, 
and compute the coefficient m for experiments 30 and 31 on the center-vent 

Boott turbine. 




464 


Chap. 14 . Turbines 


Art. 176. Theory of Reaction Turbines 

The theory of reaction turbines may be said to include two 
problems: first, given all the dimensions of a turbine and the 
head under which it works, to determine the maximum efficiency, 
and the corresponding speed, discharge, and power; and second, 
having given the head and the quantity of water, to design a 
turbine of high efficiency. This article deals only with the first 
problem, and it should be said at the outset that it cannot be 
fully solved theoretically, even for the best-conditioned wheels, 
on account of losses in foam, friction, and leakage. The investi¬ 
gation will be limited to the case of full gate, since when the gate 
is partially depressed, a loss of energy results from the sudden 
expansion of the entering water. 

The notation will be the same as that used in Chaps. 11 
and 12, and as shown in Figs. 174& and 174c; the reasoning will 
apply to both outward- and inward-flow turbines. Let r be the 
radius of the circumference where the water enters the wheel and 
r\ that of the circumference where it leaves, let u and U\ be the 
corresponding velocities of revolution; then ur\ = U\r. Let Vo 
be the absolute velocity with which the water leaves the guides 
and enters the wheel, and V its velocity of entrance relative to the 
wheel; let a be the approach angle and </> the entrance angle 
which these velocities make with the direction of u. At the exit 
circumference let V\ be the relative velocity with which the water 
leaves the guides, and v\ its absolute velocity; let ft be the exit 
angle which \\ makes with this circumference. Let a 0 , a , and ai 
be the areas of the guide orifices, the entrance, and the exit orifices 
of the wheel, respectively, measured perpendicular to the direc¬ 
tions of v 0 , V, and V x . Let do, d, and d x be the depths of these 
orifices; when the gate is fully raised, d Q becomes equal to d. 

The areas a 0 , a, a h neglecting the thickness of the guides and 
vanes, and taking the gate as fully open, have the values 

a 0 = 27rrd sina a = 277rdsin# a\ = 27rr 1 d 1 sin/3 
and since these areas are fully filled with water, 

q = v o • 27 rrd since = V • 2 ?rrd sin# = V x • 2 M 1 sin/3 (176)i 


Theory of Reaction Turbines. Art. 176 


465 


These relations, together with the formulas of the last article 
and the geometrical conditions of the parallelograms of velocities, 
include the entire theory of the reaction turbine. 

In order that the efficiency of the turbine may be as high as 
possible the water must enter tangentially to the vanes, and the 
absolute velocity of the issuing water must be as small as possible. 
The first condition will be fulfilled when u and v 0 are proportional 
to the sines of the angles <f> — a and <j>. The second will be se¬ 
cured by making U\ = V\ in the parallelogram at exit, as then 
the diagonal V \ becomes very small. Hence 

U Sin(<£ ( l ) t r (A h 7f t \ 

— = ——■ Ui=Vi ( 1 / 6)2 

S1119 


are the two conditions which should obtain in order that the 
hydraulic efficiency may be a maximum. 

Now making V\ = U\ in the third quantity of (176) 1 and 
equating it to the first, there results 

U\ _ rd simx j u_ _ r 2 d sin« 

^0 Mi sin (3 v 0 rdd sin/3 


Also making \\ = ii\ in (175)i and substituting for V 2 its value 
u 2 + Vo 2 — 2uv 0 cos« from the triangle at A between u and v 0 , 
there is found the important relation 

uv 0 coscc = gh (176)3 


which gives another condition between u and vq. The velocity 
v 0} with which the water enters, hence depends upon the speed of 
the wheel as well as upon the head h. 

Thus three equations between two unknown quantities u and 
have been deduced for the case of maximum hydraulic efficiency, 
namely, 

u _ sin(<ft ~ a) u _ r 2 d sin« ^ gh 

Vo sin (f> Vq r\ 2 d\ sin^ cos« 


If the values of the velocities u and be found from the first and 
third equations, they are 


u = 


gh sin( 4> — a) 

cos« sine p 


Vo 


gh sin (f> 


cos« sin (</> — a) 


(176), 












46G 


Chap. 14 . Turbines 


the first of which is the advantageous velocity of the circumference 
where the water enters, and the second is the absolute velocity 
with which the water leaves the guides and enters the wheel. 
In order, however, that these expressions may be correct, the 
first and second values of u/vq must also be equal, and hence 


sin (<ft — a) _ r~d sin« 
sinc/> rddi sin/3 


(176) 5 


which is the necessary relation between the dimensions and 
angles of the wheel in order that this theory may apply. 

For a turbine so constructed and running at the advantageous 
speed the theoretic hydraulic efficiency is 

V 2 2 U\ sin 2 

e — i -— = i- £ -— 

2 gh gh 

and substituting for u,\ its value in terms of u from (176)4, and 
having regard to (176) 5 , this becomes 


e = 


— tana tan^/3 
d i 



The discharge under the same conditions is q = a 0 v 0 , and lastly 
the work of the wheel per second is k = wqhe. 


The result of this investigation is that the general problem of 
investigating a given turbine cannot be solved theoretically, unless 
it be so built as to approximately satisfy the condition in (176)5. 
If this be the case, it may be discussed by the formulas deduced. Even 
then no very satisfactory conclusions can be drawn from the numerical 
values, since the formulas do not take into account the loss by friction 
and that of leakage. To determine the actual efficiency, best speed, 
and power of a given turbine, the only way is to actually test it by the 
method described in Art. 149. The above formulas are, however, 
of great value in the discussion of the design of turbines. More 
exact formulas, from a theoretical standpoint, may be derived by using 
the condition V l = n x cos/3 instead of V l = u x to determine the exit 
velocity (Art. 168), but these are very complex in form, and numeri¬ 
cal values computed from them differ but little from those found from 
the formulas here established. 


When the coefficient of discharge of a turbine is known (Art. 
175), the advantageous speed and corresponding discharge may be 






Design of Reaction Turbines. Art. 177 


467 


closely computed. For this purpose the condition = V l =q/a L is 
to be used. Inserting in this the value of q from (175) 2 and solving 
for u Y , there is found 



2gh 




,2 


a\ 


2 


i -f c 2 -b 

n 2 oo 2 



which gives the advantageous velocity of the circumference where the 
water leaves the wheel, and then by (175) 2 the discharge can be ob¬ 
tained. As an example, take the case of Holyoke test No. 275 , where 
r \ = 2 l\ inches, r = 21 J inches, h = 23.8 feet, a 0 = 2 . 066 , a = 5 . 526 , 
a-i = 1-949 square feet, a = 25 ^°, </> = 90 °, j3 = nf°. Assuming 
c = 0 . 95 , as the turbine is similar to that investigated in the last 
article, the above formula gives u x = 31.24 feet per second, which cor¬ 
responds to 130 revolutions per minute, and this agrees well with the 
actual number 138 . The efficiency found by the test at that speed was 
0 . 79 , which is a very much less value than the above theoretic formula 
gives, since this formula was derived without taking into account the 
friction losses within and without the wheel. 

Prob. 176 . For the case of the last problem r = 4.67, Y\ = 3.95, 
d = 1.01, d 1 = 1.23, h = 13.4 feet, « = 9°.5, = 119 0 , (3 = ii°. Compute 
the areas a 0 , a, a\, and the advantageous speed. Compute also the velocity 
with which the water enters the wheel. 


Art. 177. Design of Reaction Turbines 

The design of an outward- or inward-flow turbine for a given 
head and discharge includes the determination of the dimensions 
r, n, d, d h and the angles a, j3, and <j>. These may be selected in 
very many different ways, and the formulas of the last article 
furnish a guide how to make a selection so as to secure a high 
degree of efficiency. 

First, it is seen from (176) 6 that the approach angle a and the 
exit angle P should be small, but that, as in other wheels, /3 has 
a greater influence than a. However, fi must usually be greater 
for an inward-flow than for an outward-flow wheel in order to 
make the orifices of exit of sufficient size. For the entrance angle 
$ a good value is 90 °, and in this case the velocity u is always that 
due to one-half the head, as seen from (176) 4 - The radii r and n 




468 


Chap. 14 . Turbines 


should not differ too much, as then the frictional resistance of the 
flowing water and the moving wheel would be large. It is also 
seen that the efficiency is increased by making the exit depth d\ 
greater than the entrance depth d, but usually these cannot greatly 
differ, and are often taken equal. 

Secondly, it is seen that the dimensions and angles should be 
such as to satisfy the formula ( 176 ) 5 , since if this be not the case 
losses due to impact at entrance will occur which will render the 
other formulas of little value. 

As a numerical illustration let it be required to design an out¬ 
ward-flow reaction turbine which shall use 120 cubic feet per second 
under a head of 18 feet and make 100 revolutions per minute. Let 
the entrance angle be taken at 90°, then from formula ( 176) 4 the 
advantageous velocity of the inner circumference is 

u = V32.16 X 18 = 24.06 feet per second, 

and hence the inner radius of the wheel is 

r = 60 X 2 *-° 6 = 2.29S feet. 

27 T X IOO 

Now let the outer radius of the wheel be 3 feet, and also let the 
depths d and d l be equal; then from ( 176) 5 

sin£ = / M2 8y = 

tana \3.000 J 

If the approach angle a be taken as 30°, the value of the exit angle 
to satisfy this equation is 19 0 48', and from ( 176) 6 the hydraulic 
efficiency is 0.899. If, however, « be 24 0 , the value of /3 is /I 15 0 08' 
and the hydraulic efficiency is 0.941; these values of a and /? will 
hence be selected. 

• 

The depth d is to be chosen so that the given quantity of water 
may pass out of the guide orifices with the proper velocity. This 
velocity is, from ( 176 ) 4 , 

z>o = 24.06/cos 24 0 = 26.34 feet per second; 

and hence the area of the guide orifices should be 

a 0 = 120/26.34 = 4.556 square feet, 

from which the depth of the orifices and wheel is 

d = 4.556/27rr sin 24 0 = 0.776 feet. 






Guides and Vanes. Art. 178 


469 


As a check on the computations the velocities V and V h with the cor¬ 
responding areas a and a 0f may be found, and d be again determined 
in two ways. Thus, 

V = Vo sin 24 0 = 10.71 IT = u\ = ur\/r = 31.42 feet per second. 

a = 120/10.71 = 11.204 a\ = 120/31.42 = 3.820 square feet. 

d = ii.204/27!t = 0.776 di = 3.820/27rri sin /3 = 0.776 feet. 

And this completes the preliminary design, which should now be re¬ 
vised so that the several areas may not include the thickness of the 
guides and vanes (Art. 178 ). 

Although the hydraulic efficiency of this reaction turbine is 94 
percent, the practical efficiency will probably not exceed 80 percent. 
About 2 percent of the total work will be lost in axle friction. The 
losses due to the friction of the water in passing through the guides 
and vanes, together with that of the wheel revolving in water, and per¬ 
haps also a loss in leakage, will probably amount to more than one- 
tenth of the total work. All of these losses influence the advantageous 
velocity, so that a test would be likely to show that the highest effi¬ 
ciency would obtain for a speed somewhat less than 100 revolutions 
per minute. 

Prob. 177 . Design an inward-flow reaction turbine which shall use 
120 cubic feet of water per second under a head of 18 feet while making 
100 revolutions per minute, taking </> = 68°, a = io°, and /3 = 21 0 . Also 
taking <f> = 75 0 , a = 15 0 , and (d = 20°. 

Art. 178 . Guides and Vanes 

The discussions in the last two articles have neglected the 
thickness of the guides and vanes. As these, however, occupy 
a considerable space, a more correct investigation will here be 
made to take them into account. Let t be the thickness of a 
guide and n their number, h the thickness of a vane and n\ their 
number. Then the areas a 0 , a , and a Y perpendicular to the direc¬ 
tions of Vo, V, and V\ are strictly 

u 0 = ( 27 tt since — nt)d a = (27 rr sin<£ —nih)d 
ai (277T1 sin /3 — Uih) d\ 

and the expressions for the discharge in ( 176 )i are 

q = a 0 v 0 = aV = a\V 1 


470 


Chap. 14 . Turbines 


and, since V i equals u i, these give 

U\ do U CLqY 

V 0 di Vq diTi 

also, the necessary condition in ( 176) 5 becomes 

sin(9 — a) _ dpr 
sin 9 diYi 

and the greatest hydraulic efficiency of the turbine when running 
at the advantageous speed is given by 

r\ sin (9 — «) sin 2 ^/3 

e = 1 - 2 —- Y - r - - -“ 

r sin 9 cosa 

in which, of course, sin(9 — «)/ sin 9 may be replaced by its 
equivalent dor/diri. The advantageous speed is, as before, given 
by formula ( 176) 4 

To discuss a special case, let the example of the last article be 
again taken. An outward-flow turbine is to be designed to use 120 
cubic feet of water under a head of 18 feet while making 100 revo¬ 
lutions per minute, the gate being fully opened. The preliminary 
design has furnished the values r = 2.298 feet, r x = 3.000 feet, d = d x — 
0.766 feet, 9 = go°, a = 24 0 , /3 = 15 0 08'. It is now required to 
revise these so that 24 guides and 36 vanes maybe introduced. Each 
of these will be made one-half an inch thick, but on the inner circum¬ 
ference of the wheel the vanes will be thinned or rounded so as to pre¬ 
vent shock and foam that might be caused by the entering water 
impinging against their ends (see Fig. 182 c). If the radii and angles 
remain unchanged, the effect of the vanes will be to increase 
the depth of the wheel, which is now 0.702 feet wide and 0.776 feet 
deep. As these are good proportions, it will perhaps be best to keep 
the depth and the radii unchanged, and to see how the angles and the 
efficiency will be affected. 

Since the vanes are to be thinned at the inner circumference, 
the area a is unaltered and its value is simply 27 rrd sin<£. Hence 
9 remains 90° and V is unchanged. This requires that the area a 
should remain the same as before. The area d\ is also the same, as 
its value is q / u \. Accordingly the equations result 

4-556 = (2 tt r sin« — 24/) d 3.820 = (2 ttyi sin/? — 36/1) di 







Downward-flow Turbines. Art. 179 


471 


in which a and /? are alone unknown. Inserting the numerical values 
and solving, a = 28° 26' and ft = ig° 55', both being increased by 
about 4i°. The efficiency is now found to be 0.898, a decrease of 
0.043, due to the introduction of the guides and vanes. 

The efficiency may be slightly raised by making the outer depth 
di greater than the inner depth d. For instance, let d x = 0.816 
while d remains 0.776; then /3 is found to be 19 0 06', and e = 0.906. 
But another way is to thin down the vanes at the exit circumference 
and thus maintain the full area a x with a small angle fi. If this be 
done in the present case d x may be kept at 0.776 feet, ft be reduced 
to about 16 0 , and the efficiency will then be about 0.92 or 0.93. 

No particular curve for the guides and vanes is required, but it 
must be such as to be tangent to the circumferences at the designated 
angles. The area between two vanes on any cross-section normal 
to the direction of the velocity should also not be greater than the area 
at entrance; in order to secure this vanes are frequently made much 
thicker at the middle than at the ends (see Fig. 182 c). 

Prob. 178 . Find the advantageous speed and the probable discharge 
and power of the turbine designed above when under a head of 50 feet. 


Art. 179 . Downward-flow Turbines 

Downward- or parallel-flow turbines are those in which the 
water passes through the wheel without changing its distance from 
the axis of revolu- 


G 


w 


r 


7 .7 7;; 


5 



tion. In Fig. 179 a 
is a semi-vertical 
section of the guide 
and wheel passages, 

and also a develop- Fig 179a 

ment of a portion 
of a cylindrical section showing the inner arrangement. The 
formula for the discharge can be adapted to this by making 
ui = u. In this turbine there is no action of centrifugal force, 
so that the relative exit velocity Vi is equal to the relative en¬ 
trance velocity V. 

The great advantage of this form of turbine is that it can be 
set some distance above the tail race and still obtain the power 




















472 


Chap. 14. Turbines 


i 


due to the total fall. This distance cannot exceed 34 feet, the 
height of the water barometer, and usually it does not exceed 

25 feet. Fig. 179 Z> shows in a dia¬ 
grammatic way a cross-section of the 
penstock P, the guide passages G, 
the wheel IF, and the air-tight draft 
tube T, from which the water es¬ 
capes by a gate E to the tail race. 
The pressure-head Hi on the exit 
orifice is here negative, so that the 
air pressure equivalent to this head 
is added to the water pressure in 
the penstock, and hence the discharge 
through the guides occurs as if the 
wheel were set at the level of the tail 
race. Strictly speaking, a vacuum, 
more or less complete, is formed just 
below the wheel into which the water 
drops with a low absolute velocity, 
having surrendered to the wheel 
nearly all its energy. Draft tubes 
are also often used with inward-flow turbines when these are set 
above the tail race. 

Let li be the total head between the water levels in the head and 
tail races, 7/ 0 the depth of the entrance orifices of the wheel below the 
upper level, h x the vertical height of the wheel, and h 2 the height of the 
exit orifices above the tail race; so that h = // 0 -h 7 ?i+/^. Let H and 
H x be the heads which measure the absolute pressures at the entrance 
and exit orifice of the wheel, and 7 / a the height of the water barometer. 
Let t' 0 be the absolute velocity with which the water leaves the guides 
and enters the vanes, and V and V 1 the relative velocities at entrance 
and exit. Then from the theorem of energy in steady flow (Art. 31 ), 

Vo 2 = 2 g (h a + h 0 — H) 

VV = V 2 + 2 g(hi + H-Hi) 

Adding these two equations there results 

Vo 2 — V 2 + F1 2 = 2 g (ho + hi + h a — Hi) 


































































Downward-flow Turbines. Art. 179 


473 


But h a — H x is equal to h 2 , and hence 

Vo 2 -V 2 -\- V \ 2 = 2 gh 

This formula is the same as ( 175 )i if u be made equal to u h and hence 
all the formulas of the last three articles apply to the downward-flow 
reaction turbine by making equal the velocities u and u h as also the 
radii r and r x . 

Let r be the mean radius and u the mean velocity of the entrance 
and exit orifices of the wheel, let d be the width of the entrance orifices 
and d x that of the exit orifices. Let a be the approach angle which the 
direction of the entering water makes with that of the velocity u, 
or the angle which the guides make with the upper plane of the wheel 
(Fig. 179 a); let <f> be the entrance angle which the vanes make with 
that plane, and /3 the acute exit angle which they make with the lower 
plane. Then the values of the advantageous velocity u and the enter¬ 
ing velocity v Q are 

Usin (f-« ; , o 
\ cos«sm<p 

and the necessary relation between the angles of the vanes and the 
dimensions of the wheel is 

sin (<f> — «) _ d sina do 

sine j> di sin /3 a\ 

while the hydraulic efficiency of the turbine is 

ao sin 2 d , . i n 

e = i — 2 —-= i —-tan«t.an^p 

d\ cosa di 

To these equations is to be added the condition that the pressure- 
head Hi cannot be less than that of a vacuum, and on account of air 
leakage it must be practically greater; thus 

H\>o and h<i<h a 

that is, the height of the wheel orifices above the tail race must be 
less than the height of the water barometer. 

As an example of design, let </> = 90° and a = 30°. Then u = 
Jgh , or the velocity due to one-half the head; and v Q = V-f gh, or a 
' velocity due to two-thirds of the head. From the above formulas, 
taking d x = f d, the value of /? is 22 0 38' and the efficiency is found to. 
be 0.92. This value will be lowered by the introduction of guides and. 


gh sin cf> 


cos a sin (</> — a) 










474 


Chap. 14. Turbines 


vanes, as well as by friction, so that perhaps not more than 0.80 will 
be obtained in practice. 

Prob. 179 . A downward-flow turbine with draft tube has its exit 
orifices 7.5 feet above the level of the tail race, and it uses 87 cubic feet 
of water per second under a head of 25 feet. What horse-power will this 
turbine deliver when its efficiency, as measured by the friction brake, is 76 
percent ? 


Art. 180 . Impulse Turbines 


Whenever a turbine is so arranged that the channels between 
the vanes are not fully filled with water, it ceases to act as a 
reaction turbine and becomes an impulse turbine. A turbine 
set above the level of the tail race becomes an impulse turbine 
when the gate is partially lowered, unless the gates are arranged 
so as to cover the exit orifices instead of being, as usual, in front 
of the entrance orifices. 

The velocity with which the water leaves the guides in an 
impulse turbine is simply sVg/zo, where // 0 is the head on the 
guide orifices. The rules and formulas in Art. 168 apply in all 
respects, and for a well-designed wheel the entrance angle <f> is 
double the approach angle «, the advantageous speed and corre¬ 
sponding hydraulic efficiency are 


u = 



e = 1 


/ Vi sin V 
\ r cos a J 


while the discharge is q = a 0 ^/ 2 gh 0 , and the work of the turbine 
per second is k = wqJi 0 e. 

As an example, suppose that the reaction turbine designed 
in Art. 177 were to act as an impulse turbine, the angles « and 
/3 remaining at 24° and 15 0 08', and the radii r and r x being 2.298 
and 3.000 feet. It would then be necessary that 4 > should be 
48° instead of 90° in order to secure the best results. Under a 
head of 18 feet the velocity of flow from the guides would be 
34.02 feet per second instead of 26.34. The velocity of the inner 
circumference would be 18.63 feet P er second instead of 24.06, 
so that the number of revolutions per minute would be about 77 
instead of 100. The efficiency would be 0.96, or almost exactly 







Impulse Turbines. Art. 180 


475 


the same as before. If, however, the angle <£ were to remain 90°, 
the efficiency of the turbine would be materially lowered, since 
then the water could not enter tangentially upon the vanes, 
and a loss in energy of the entering water due to the impact 
would necessarily result. 

Impulse turbines revolve more slowly than reaction turbines 
under the same head, but the relative entrance velocity V is 
greater, and hence more energy is liable to be spent in shock and 
foam. In impulse turbines the entrance angle <ft should be double 
the approach angle «, but in reaction turbines it is often greater 
than 3 a, and its value depends upon the exit angle / 3 ; hence the 
vanes in impulse turbines are of sharper curvature for the same 
values of « and / 3 . In impulse turbines the efficiency is not low¬ 
ered by a partial closing of the gates, whereas the sudden enlarge¬ 
ment of section causes a material loss in reaction turbines. The 
advantageous speed of an impulse turbine remains the same for 
all positions of the gate, but with a reaction turbine it is very much 
slower at part gate than at full gate. For many kinds of machin¬ 
ery it is important to maintain a constant speed for different 
amounts of power, and with a reaction turbine this can only be 
done by a great loss in efficiency. When the water supply is low, 
the impulse turbine hence has a marked advantage in efficiency. 
A further merit of the impulse turbine is that it may be arranged 
so that water enters only through a part of the guides, while this 
is impossible in reaction turbines. On the other hand, reaction 
turbines can be set below the level of the tail race or above it, 
using a draft tube in the latter case, and still secure the power 
due to the total fall, whereas an impulse turbine must always 
be set above the tail-race level and loses all the fall between 
that level and the guide orifices. 

Prob. 180 a. Compare the advantageous speeds of impulse and reaction 
turbines when the velocity of the water issuing from the guide orifices is 
the same. 

Prob. 1806 . Design an outward-flow impulse turbine which shall use 
120 cubic feet of water per second under a head of 18 feet and make 100 rev¬ 
olutions per minute. Compare the dimensions and angles with those of 
the reaction turbine designed for the same data in Art. 177 . 


476 


Chap. 14. Turbines 


Art. 181 . Special Devices 

Many devices to increase the efficiency of reaction turbines, 
particularly at part gate, have been proposed. In the Fourney- 
ron turbine a common plan is to divide the wheel into three parts 
by horizontal partitions between the vanes, so that these are 
completely filled with water when the gate is either one-third 
or two-thirds closed (see Fig. 182 d). The surface exposed to 
friction is thus, however, materially increased at full gate. 

The Boyden diffuser is another device used with outward-flow 
reaction turbines. This consists of a fixed wooden annular 
frame D placed around the wheel W, through which the water 
must pass after exit from the wheel. Its width is about four or 

five times that of the wheel, and 
at the outer end its depth becomes 
about double that of the wheel. 
The effect of this is like a draft 
tube, and although the absolute 
velocity of the water when issuing 
from the wheel is greater than be¬ 
fore, the absolute velocity of the 
water coming out of the diffuser is less,, and hence a greater 
amount of energy is imparted to the turbine. It has been 
shown above that the efficiency of a reaction turbine is increased 
by making the exit depth greater than the entrance depth d, 
and the fixed diffuser produces the same result. By the use of 
this diffuser Boyden increased the efficiency of the Fourneyron 
reaction turbine several percent. 

The pneumatic turbine of Girard was devised to overcome 
the loss in reaction turbines due to a partial closing of the gate. 
The turbine was inclosed in a kind of bell into which air could 
be pumped, thus lowering the tail-water level around the wheel. 
At part gate this pump is put into action, and as a consequence 
the air is admitted into the wheel, and the water flowing through 
it does not fill the spaces between the vanes. Hence the action 
becomes like that of an impulse turbine, and the full efficiency 
is maintained, although power is lost in compressing the air. * 



Fig. 181. 































The Niagara Turbines. Art. 182 


477 


At a high stage of the stream, when water flows to waste over 
the dam, backwater usually lessens the available fall and power. 
To increase that fall and power, Herschel in 1908 devised and 
tested at the Holyoke Testing Flume the plan of connecting the 
lower end of the turbine draft tube to a chamber wherein a partial 
vacuum is produced by causing part of the waste to flow through 
a tube shaped like the Venturi meter, suitable connections being 
made between the specially designed throat of the tube and the 
vacuum chamber. This device, called “ the fall increaser,” * gives 
greater available power at high water stages, since the vacuum 
head hi is added to the head h between the upper and lower water 
levels, and since the discharge through the turbine is also increased. 

The screw turbine consists of one or two turns of a helicoidal sur¬ 
face around a vertical shaft, the screw being inclosed in a cylindrical 
case. At a point of entrance the downward pressure of the water can 
be resolved into two components, a relative velocity V parallel to the 
surface and a horizontal velocity u which corresponds to the velocity of 
the wheel. At the point of exit it can be resolved in like manner into 
V 1 and u L . But, as in other cases, the condition for high efficiency is 
iii ~ Fi, and since the water moves parallel to the axis, u x — u. 
Applying the general formula of Art. 175 , it is seen that this can only 
occur when the head h is zero or when the velocity u is infinite. The 
screw turbine is hence like a reaction wheel, and high efficiency can 
never practically be obtained. 

Prob. 181 . Consult Riihlmann’s Maschinenlehre, vol. 1, pp. 360-425, 
and describe a scheme for “ventilating” a turbine in order to increase its 
efficiency. 

Art. 182 . The Niagara Turbines 

A number of turbines have been installed at Niagara Falls, 
N.Y., for the utilization of a portion of the power of the great 
falls. Those to be here briefly described are the ten large wheels 
designed by Faesch and Picard, of Geneva, Switzerland, and 
erected from 1894 to 1900 for the Niagara Falls Power Company. 
The entire plant is to include twenty-one twin outward-flow 
reaction turbines, each of about 5000 horse-power. It is located 


* Harvard Engineering Journal, June, 1908. 


478 


Chap. 14. Turbines 


about 11 miles above the 
American fall, where a canal 
leads water from the river 
to the wheel pit. The water 
is carried down the pit 
through steel penstocks to 
the turbines, which are 
placed 136 feet below the 
water level in the canal. 
After passing through the 
wheels the waste water is 
conveyed to the river below 
the American fall by a tun¬ 
nel 7000 feet long.* 

Fig. 182 a shows a cross- 
section of the wheel-pit, 
with an end view of a pen¬ 
stock, wheel case, and shaft. 
Fig. 1826 exhibits part of a 
longitudinal section of the 
wheel pit and a side view 
of two of the penstocks, 
with the inclosing cases and 
shafts of the turbines. These 
figures show a rock-surface 
wheel pit, but this surface 
was later protected by a 
brick lining having a thick¬ 
ness of about 15 inches. 
The width of the wheel pit 
is 20 feet at the top and 16 
feet at the bottom, and the 
cylindrical penstock is 7J 
feet in diameter. The shaft 
of the turbine is a steel tube 



* Engineering News, 1892, vol. 27, p. 74, and 1893, vol. 29, p. 294. 





















































































































































































The Niagara Turbines. Art. 182 479 



Fig. 1S2Z», 





















































































































































































































































































































































































480 


Chap. 14. Turbines 


38 inches in diameter, built in three sections, and connected by 
short solid steel shafts n inches in diameter which revolve in 

bearings. At the . 
top of each shaft 
is a dynamo for 
generating the 
electric power. 

In Fig. 182 c 
is shown a vertical 
section of the 
lower part of the 
penstock, shaft, 
and twin wheels. 
The water fills the 
casing around the 
shaft, passes both 
upward and 
downward to the 
guide passages, 
marked G, through 
which it enters 
the two wheels, 
causes them to 
revolve, and then drops down to the tail race at the entrance 
to the tunnel, which carries it away to the river. The gate for 
regulating the discharge is seen upon the outside of the wheels. 

Fig. 182 d gives a larger vertical section of the lower wheel 
with the guides, shaft, and connecting members. The guide 
passages, marked G , and the wheel passages, marked W, are 
triple, so that the latter may be filled not only at full gate, but 
also when it is one-third or two-thirds opened, thus avoiding the 
loss of energy due to sudden enlargement of the flowing stream. 
The two horizontal partitions in the wheel are also advantageous 
in strengthening it. The inner radius of the wheel is 31J inches 
and the outer radius is 37J inches, while the depth is about 12 
inches. In this figure the gates are represented as closed. 



Fig. 182c. 














































































































































































































The Niagara Turbines. Art. 182 


481 


In Fig. 182 e is shown a half-plan of one of the wheels, on a 
part of which are seen the guides and vanes, there being 36 of 



the former and 32 of the latter. The value of the approach angle 
a is 19 0 06', the mean value of the entrance angle is no° 40', 
and the exit angle /3 is 13 0 iyi'. Although the water on leaving 



the wheel is discharged into the air, the very small annular space 
between the guides and vanes, together with the decreasing area 































































































































































































































482 


Chap. 14. Turbines 


between the vanes from the entrance to the exit orifices, insures 
that the wheels act like reaction turbines for the three positions 
of the gates corresponding to the three horizontal stages. 

The average discharge through one of these twin turbines is 
about 430 cubic feet per second, and the theoretic power due to 
this discharge is 6645 horse-powers. Hence if 5000 horse-powers 
be utilized, the efficiency is 75.2 percent. Under this discharge 
the mean velocity in the penstock is nearly 10 feet per second, 
but the loss of head due to friction in the penstock will be but a 
small fraction of a foot. The pressure-head in the wheel case is 
then practically that due to the actual static head, or closely 
141I feet upon the lower and 130 feet upon the upper wheel. 
Although the penstock is smaller in section than generally 
thought necessary for such a large discharge, the loss of head 
that occurs in it is insignificant; and it will be seen in Fig. 
182 a to be connected with the head canal and with the wheel 
case by easy curves, and that its section is enlarged in making 
these approaches. 

A test of one of these wheels, made in 1895, showed that 5498 
electrical horse-powers were generated by an expenditure of 447.2 
cubic feet of water per second under a head of 135.1 feet. The effi¬ 
ciency of the dynamo being 97 percent, the efficiency of the wheel 
and approaches was 82f percent. The water was measured, 
when entering the penstock, by a current meter of the kind illus¬ 
trated in Art. 40 . 

From formula ( 176)4 the advantageous velocity of the inner 
circumference of the upper wheel, taking h = 130! feet, is found 
to be 68.88 feet per second, and that for the lower wheel, taking 
. h = 141J feet, is found to be 71.73 feet per second. Perhaps 
the mean of these, or 70.31 feet per second, closely corresponds 
with the advantageous velocity for the two combined. The 
number of revolutions per minute for the condition of maximum 
efficiency is then closely 250. The absolute velocity of the water 
when entering the wheel is about 66 feet per second, so that the 
pressure-head in the guide passages of the upper wheel is nearly 
66 feet. The mean absolute velocity of the water when leaving 


The Niagara Turbines. Art. 182 


483 


the wheels is about 19 feet per second, so that the loss due to this 
is only about 4 percent of the total head. 

The weight of the dynamo, shaft, and turbine is balanced, 
when the wheels are in motion, by the upward pressure of the 
water in the wheel case on a piston placed above the upper wheel. 
The upper disk containing the guides is, for this purpose, per¬ 
forated, so that the water pressure can be transmitted through 
it. In Fig. 182 c these perforations can be seen, and the balancing 
piston is marked B. The lower disk, on the other hand, is solid, 
and the weight of the water upon it is carried by inclined rods 
upward to the wheel case, which together with the penstock is 
supported upon several girders. At the upper end of the shaft is 
a thrust bearing to receive the excess of vertical pressure, which 
may be either upward or downward under different conditions 
of power and speed. 

A governor is provided for the regulation of the speed, and this 
is located on the surface near the dynamo. It is of the centrifugal- 
ball type, and so connected with the main shaft and the turbine 
gates that the latter are partially closed whenever from any 
cause the speed increases. These gates are so set that the orifices 
of the upper and lower wheels are not simultaneously closed, one 
gate being in advance of the other by about the width of one 
division stage. The revolving field magnets of the dynamo also 
serve as a fly-wheel for equalizing the speed. With this method 
of regulation it is insured that the speed cannot increase more than 
3 or 4 percent when 25 percent of the work is suddenly removed. 

The above description refers to the ten turbines in wheel pit 
No. 1 . The illustrations are those of the wheels called units 1 , 2 , and 3 , 
which are installed in 1894 and 1895. Units 4 to 10 inclusive, installed 
in 1898-1900, are of the same type except that both the penstock and 
wheel case have cast-iron ribs on their sides which rest on massive 
castings built into the masonry of the side walls. This arrangement 
dispenses with the supporting girders shown in Figs. 182 u- 182 r, 
and gives much greater rigidity to both penstocks and wheels. 

The excavation of a new wheel pit, called No. 2 , was begun in 1896, 
and the installation of units 11-21 was completed in 1903. These 


484 


Chap. 14 . Turbines 


wheels have penstocks and shafting similar to those of units 1-10> 
but the wheels are of the Jonval type, the flow being inward and down¬ 
ward. The wheel case has the form of a flattened sphere, the water 
entering from one side and passing through the guides to a single 
turbine 64 inches in diameter and 23.5 inches deep. After leaving the 
w r heel, the water passes to two draft tubes, each about 58 inches in 
diameter, and is discharged near the invert of the tail race at an angle 
of 45 0 to the horizontal axis of the wheel pit. The wheel case is sup¬ 
ported on these two draft tubes as on two legs, while the penstock 
is supported on iron lugs in the same way as those of units 4 - 10 . 
By these draft tubes the head on the w T heel is increased to 144 feet, 
this being the difference from the water level in the head race to that 
in the tail race. The balancing pistons are below the wheels, and are 
supported from an independent pipe instead of from the penstock. 
Each shaft is also supplied w r ith an oil step-bearing, which is designed 
to' support, if necessary, the entire revolving weight at the normal 
speed of 250 revolutions per minute. 

Prob. 182a. Compute the hydraulic efficiency of the turbines described 
above. Compute the velocity 7> 0 with which the water enters the lower wheel 
and the velocity Vi with which it leaves the same when the speed is 250 
revolutions per minute. 

Prob. 1825. Compute the efficiency of a reaction wheel under a head of 
3.5 meters when the radius of the exit orifices is 0.64 meters, the coefficient 
of velocity 0.95, and the number of revolutions per minute is 130. 

Prob. 182c. Design an outward-flow reaction turbine which shall use 8 
cubic meters of water per second uncfer a head of 12.4 meters, taking the en¬ 
trance angle <£ as 90°. 

Prob. 182 d. A dynamo delivering 4100 kilowatts has an efficiency of 97.5 
percent, while the efficiency of the turbine is 81.3 percent and that of the 
approaches to the turbine is 99.7 percent. The turbine is of the Jonval type, 
and the difference between the levels of head and tail race is 14.4 meters. 
How many cubic meters of water are used per second ? 

Prob. 182c. Consult engineering periodicals and describe other large 
power plants for the development of electrical energy which have been in¬ 
stalled at Niagara Falls, especially that of the Canadian Niagara Power 
Company and that of the Ontario Power Company. 


General Principles. Art. 183 


485 


CHAPTER 15 
NAVAL HYDROMECHANICS 
Art. 183 . General Principles 

In this chapter is to be discussed in a brief and elementary 
manner the subject of the resistance of water to the motion of 
vessels, and the general hydrodynamic principles relating to their 
propulsion. The water may be at rest and the vessel in motion, 
or both may be in motion as in the case of a boat going up or 
down a river. In either event the velocity of the vessel relative 
to the water need only be considered, and this will be called v. 
The simplest method of propulsion is by the oar or paddle; then 
come the paddle wheel, and the jet and screw propellers. The 
action of the wind upon sails will not be here discussed, as it is 
outside of the scope of this book. 

The unit of linear measure used on the ocean is generally 
the nautical mile, while one nautical mile per hour is called a 
knot. One nautical mile is about 6080 feet, so that knots may 
be transformed into feet per second by multiplying by 1.69, and 
feet per second may be transformed into knots by multiplying 
by 0.592. On rivers the speed is estimated in statute miles per 
hour, and the corresponding multipliers will be 1.47 and 0.682. 
One kilometer per hour equals 0.621 miles per hour or 0.91 feet 
per second. On the ocean the weight of a cubic foot of water is 
to be taken as about 64 pounds (it is often used as 64.32 pounds, 
so that the numerical value is the same as 2 g), and in rivers at 
62.5 pounds. 

The speed of a ship at sea was formerly roughly measured 
by observations with the log, which is a triangular piece of wood 
attached to a cord which is divided by tags into lengths of about 
50! feet. The log being thrown into the water, it remains sta- 


486 


Chap. 15. Naval Hydromechanics 


tionary, the ship moves away from it, and the number of tags 
run out in half a minute is counted; this number is the same as 
the number of knots per hour at which the ship is moving, since 
5of feet is the same part of a knot that a half minute is of an hour. 
The patent log, which is a small self-recording current meter, 
drawn in the water behind the ship, is, however, now generally 
used, this being rated at intervals (Art. 40 ). In experimental 
work more accurate methods of measuring the velocity are neces¬ 
sary, and for this purpose the boat may run between buoys 
whose distance apart has been found by triangulation from meas¬ 
ured bases on shore. 

The Pitot tube has recently been applied to the determina¬ 
tion of the velocity of a ship through the water. By the use in 
connection with this tube of a recording mechanism similar to 
that described in Art. 38 for the Venturi meter it would seem 
possible to automatically record on dials both the speed through 
the water as well as the total number of miles passed over. By 
the use of a chart an autographic record of variations in the speed 
could also be kept. Practical difficulties in the way of keeping 
the mouths of the Pitot tubes free from obstructions have already 
been to a certain extent overcome.* 

When a boat or ship is to be propelled through water, the 
resistances to be overcome increase with its velocity, and conse¬ 
quently, as in railroad trains, a practical limit of speed is soon 
attained. These resistances consist of three kinds: the dynamic 
pressure caused by the relative velocity of the boat and the water, 
the frictional resistance of the surface of the boat, and the wave 
resistance. The first of these can be entirely overcome, as in¬ 
dicated in Art. 155 , by giving to the boat a “fair” form; that is, 
such a form that the dynamic pressure of the impulse near the 
bow is balanced by that of the reaction of the water as it closes 
in around the stern. It will be supposed in the following pages 
that the boat has this form, and hence this first resistance need 
not be further considered. The second and third sources of 
resistance will be discussed later. 


* Engineering News, May 4, 1911. 


Frictional Resistances. Art. 184 


487 


The total force of resistance which exists when a vessel is 
propelled with the velocity v can be ascertained by drawing 
it in tow at the same velocity, and placing on the tow line a dy¬ 
namometer to register the tension. An experiment by Froude 
on the Greyhound, a steamer of 1157 tons, gave for the total 
resistance the following figures : * 

Speed in knots, 4 6 8 10 12 

Resistance in tons, 0.6 1.4 2.5 4.7 9.0 

which show that at low speeds the resistance varies about as 
the square of the velocity, and at higher speeds in a faster ratio. 
For speeds of 15 to 25 knots, the usual velocity of ocean steamers, 
the law of resistance is not so well known, but as an approxima¬ 
tion it is usually taken as varying with the square of the velocity. 

Prob. 183 . What horse-power was expended in the above test of the Grey¬ 
hound when the speed was 12 knots per hour? 

Art. 184 . Frictional Resistances 

When a stream or jet moves over a surface, its velocity is 
retarded by the frictional resistances, or if the velocity be main¬ 
tained uniform, a constant force is overcome. In pipes, conduits, 
and channels of uniform section the velocity is uniform, and con¬ 
sequently each square foot of the surface or bed exerts a constant 
resisting force, the intensity of which will now be approximately 
computed. This resistance will be the same as the force required 
to move the same surface in still water, and hence the results 
will be directly applicable to the propulsion of ships. 

Let F be the force of frictional resistance per square foot of 
surface of the bed of a channel, p its wetted perimeter, l its length, 
h its fall in that length, a the area of its cross-section, and v the 
mean velocity of flow. The force of friction over the entire sur¬ 
face then is Fpl, and the work per second lost in friction is 
Fplv. The work done by the water per second is Wh or wavh. 
Equating these two expressions for the work, there results 

F = w(a/p) ( h/l ) = wrs 

* Thearle’s Theoretical Naval Architecture (London, 1876), p. 347. 


488 


Chap. 15. Naval Hydromechanics 


in which r is the hydraulic radius and s the slope of the water 
surface. Now inserting for rs its value from formula ( 113 ), 
there results p _ wv 2 / c 2 


in which w is the weight of a cubic foot of water and c is the co¬ 
efficient in the Chezy formula, the values of which are given in 
Chap. 9 and the accompanying tables. Inasmuch as the 
velocities along the bed of a channel are somewhat less than the 
mean velocity v, the values of F thus determined will probably 
be slightly greater than the actual resistance. 

For smooth iron pipes the following are computed values 
of the frictional resistance in pounds per square foot of surface: 


Velocity, feet per second = 2 

4 6 

10 

15 

for 1 foot diameter F = 0.023 

0.080 0.17 

0.43 

0.92 

for 4 feet diameter F = 0.015 

0.053 0.11 

0.28 

o -59 


These figures indicate that the resistance is subject to much 
variation in pipes of different diameters; it is not easy to con¬ 
clude from them, or from formula ( 113 ), what the force of re¬ 
sistance is for plane surfaces over which water is moving. 

Experiments made by moving flat plates in still water so 
that the direction of motion coincides with the plane of the sur¬ 
face have furnished conclusions regarding the laws of fluid fric¬ 
tion similar to those deduced from the flow of water in pipes. It 
is found that the total resistance is approximately proportional 
to the area of the surface, and approximately proportional to 
the square of the velocity. Accordingly the force of resistance 
per square foot may be written 


F =fv\ ( 184 ) 

in which v is the velocity in feet per second and f is a number 
depending upon the nature of the surface. The following are 
average values of / for large surfaces, as given by Unwin : * 


Varnished surface, 

Painted and planed plank, 
Surface of iron ships, 

Fine sand surface, 

New well-painted iron plate, 


/ = 0.00250 
/ = 000339 
/ = 0.00351 
/ = 0.00405 
/ = 0.00473 


* Encyclopedia Britannica, 9th Ed., vol. 12, p. 483; 


nth Ed., vol. 14, p. 57. 


Frictional Resistances. Art. 184 


489 


Undoubtedly the value of / is subject to variations with the 
velocity, but the experiments on record are so few that the law 
and extent of its variation cannot be formulated. It should, 
however, be remarked that the formulas and constants here given 
do not apply to low velocities, for the reasons given in Art. 124 . 
At the same time they are only approximately applicable to high 
velocities. A low velocity of a body moving in an unlimited 
stream may be regarded as i foot per second or less, a high veloc¬ 
ity as 25 or 30 feet per second. 

It may be noted that the above-mentioned experiments indicate 
that the value of F is greater for small surfaces than for large ones. 
For instance, a varnished board 50 feet long gave / = 0.00250, while 
one 20 feet long gave/ = 0.00278, and one 8 feet long gave/ = 0.00325, 
the motion being in all cases in the direction of the length. The re¬ 
sistance is the same whatever be the depth of immersion, for the fric¬ 
tion is uninfluenced by the intensity of the static pressure. This is 
proved by the circumstance that the flow of water in a pipe is found to 
depend only upon the head on the outlet end, and not upon the pres¬ 
sure-heads along its length. 

The frictional resistance of a boat or ship may be roughly esti¬ 
mated by taking 0.004s; 2 and multiplying it by the immersed area. For 
instance, if this area be 8000 square feet, the frictional resistance at 
a velocity of 10 feet per second is 3200 pounds, but at a velocity of 20 
feet per second it is 12 800 pounds; the horse-powers needed to over¬ 
come these resistances are 58 and 464, respectively. To these must 
be added the power necessary to overcome the friction of the air and 
that wasted in the production of waves. 

The above discussion refers to the case of boats moving in the ocean 
and lakes or in a stream of large width and depth. In a canal the re¬ 
sistance is much greater, and it depends upon the ratio of the cross- 
section of the canal to that of the immersed portion of the boat. It de¬ 
pends also on the depth of the water. The “drag” of a ship in 
shoal water is very pronounced. . For some experiments on the suc¬ 
tion of vessels consult.* When the width of the canal is about five 
times that of the boat and the area of its cross-section about seven 
times that of the boat, the resistance is but slightly greater than in an 

* Transactions American Society of Naval Architects and Marine 
Engineers, vol. 17, 1909. 


490 


Chap. 15. Naval Hydromechanics 


unlimited stream. For smaller ratios the resistance rapidly increases, 
and when two boats pass each other in a small canal, the utmost 
power of the horses may be severely taxed. The reason for this in¬ 
creased resistance appears to be largely due to the fact that the 
velocity of the water relative to the boat increases with the diminu¬ 
tion of the cross-section of the canal. Thus, if a and A be the areas 
of the cross-section of the canal and of the immersed part of the 
boat, the effective area of the water cross-section is a — A, and the 
water flowing backward through this area must have a higher rela¬ 
tive velocity as A increases. The value of F given by formula ( 184 ) 
is accordingly increased to fv 2 /(i — (A/a)) 2 . 

Prob. 184 a. What horse-power is required to overcome the frictional re¬ 
sistance of a boat moving at the rate of g knots per hour when the area of its 
immersed surface is 320 square feet? 

Prob. 184 b. A canal has a cross-section of 360 square feet, while that of 
a canal boat is 60 square feet. Show that when two boats pass each other, 
the resistance of each is increased about 60 percent. 

Art. 185 . Work Required for Propulsion 

When a boat or ship moves through still water with a velocity 
v, it must overcome the pressure due to impulse of the water and 
the resistance due to the friction of its surface on the water and 
air. If the surface be properly curved, there is no resultant 
pressure due to impulse, as shown in Art. 155 . The resistance 
caused by friction of the immersed surface on the water can be 
estimated, as explained above. If A be the area of this surface 
in square feet, the work per second required to overcome this 


resistance is 


( 185 ) 


k = AFv = fAv* 


The work, and hence the horse-power, required to move a boat 
accordingly varies approximately as the cube of its velocity. By 
the help of the values of / given in the last article an approxi¬ 
mate estimate of the work can be made for particular cases. 
The resistance of the air, which in practice must be considered, 
will be here neglected. 

To illustrate this law let it be required to find how many tons 
of coal will be used by a steamer in making a trip of 3000 miles 
in 6 days, when it is known that 800 tons are used in making 


Work Required for Propulsion. Art. 185 491 

the trip in io days. As the power used is proportional to the 
amount of coal, and as the distances traveled per day in the two 
cases are 500 miles and 300 miles, the law gives r/480 = (5/3) 3 , 
whence T = 2220 tons. By the increased speed the expense for 
fuel is increased 277 percent, while the time is reduced 40 per¬ 
cent. If the value of wages, maintenance, interest, etc., saved 
on account of the reduction in time, will balance the extra expense 
for fuel, the increased speed is profitable. That such a compensa¬ 
tion occurs in many instances is apparent from the constant efforts 
to reduce the time of trips of passenger steamers. 

When a boat moves with the velocity v in a current which has 
a velocity u in the same direction, the velocity of the boat relative 
to the water is z; — u, and the resistance is proportional to (v — u) 2 
and the work to (y — u) 3 . If the boat moves in the opposite 
direction to the current, the relative velocity is v -f u , and of 
course v must be greater than u or no progress would be made. 
In all cases of the application of the formulas of this article and 
the last, v is to be taken as the velocity of the boat relative to the 
water. 

Another source of resistance to the motion of boats and ships is 
the production of waves. This is due in part to a different level of the 
water surface along the sides of the ship due to the variation in static 
pressure caused by the velocity, and in part to other causes. It is 
plain that waves, eddies, and foam cause energy to be dissipated in heat, 
and that thus a portion of the work furnished by the engines of the 
boat is lost. This source of loss is supposed to consume from 10 to 40 
percent of the total work, and it is known to increase with the ve¬ 
locity. On account of the uncertainty regarding this resistance, as 
well as those due to the friction of the water and air, practical compu¬ 
tations on the power required to move boats at given velocities can 
only be expected to furnish approximate results. 

The investigations of Rankine on this difficult subject led to the 
conclusion announced in 1858 in the anagram (20 a, 4 b, 6c, 9 d, 34c, Sf, 
4 g, 16/z, 10 i, 5/, 3m, 15 n, 14 0, 4p, 3?, 14/3 i3h 25/, 411, 2v, 2w, ix, 4y)* 
The meaning of this anagram was published in 1861 : “The resistance 
of a sharp-ended ship exceeds the resistance of a current of water of 

* Philosophical Magazine, September, 1858. 


492 


Chap. 15. Naval Hydromechanics 


the same velocity in a channel of the same length and mean girth by a 
quantity proportional to the square of the greatest breadth divided by 
the square of the length of the bow and stern. ” 

Prob. 185 . Compute the horse-power required to maintain a velocity of 
18 knots per hour, taking A = 7473 square feet and / = 0.004. 


Art. 186 . The Jet Propeller 


The method of jet propulsion consists in allowing water to 
enter the boat and acquire its velocity, and then to eject it back¬ 
wards at the stern by means of a pump. The reaction thus pro¬ 
duced propels the boat forward. To investigate’the efficiency 
of this method, let W be the weight of water ejected per second, 
V its velocity relative to the boat, and v the velocity of the boat 
itself. The absolute velocity of the issuing water is then V — v, 
and it is plain without further discussion that the maximum 
efficiency will be obtained when this is o, or when V = v, as then 
there will be no energy remaining in the water which is propelled 
backward. It is, however, to be shown that this condition can 
never be realized and that the efficiency of jet propulsion is low. 


The effective work which is exerted on the boat by the reaction 
of the issuing water is /Tr N 

k = w WzAi 

g 

and the work lost in the absolute velocity of the water is 


k ' = w^A£ 

2g 

The sum of these is the total theoretic work, or 

V 2 — v 2 


K = W 


2 g 


Therefore the efficiency of jet propulsion is expressed by 


— — 2 v 
K~V + v 


This becomes equal to unity when v = V as before indicated, but 
then it is seen that the work k becomes o unless W is infinite. The 
value of W is waV, if a be the area of the orifices through which 






Paddle Wheels. Art. 187 


493 


the water is ejected; and hence in order to make e unity and at 
the same time perform work it is necessary that either V or a 
should be infinity. The jet propeller is therefore like a reaction 
wheel (Art. 172 ), and it is seen upon comparison that the formula 
for efficiency is the same in the two cases. 

By equating the above value of the useful work to that es¬ 
tablished in the last article there is found 

fgA v 2 = wa V{V — v) 

and if this be solved for V, and the resulting value be substituted 
in the formula for e, it reduces to 


3 + Vi -\r{^fgA/wa) 

which again shows that e approaches unity as the ratio of a to A 
increases. The area of the orifices of discharge must hence be 
very large in order to realize both high power and high efficiency. 
For this reason the propulsion of vessels by this method has not 
proved economical, although in the case of the boat Waterwitch, 
built in England about i860, a fair speed was attained. In nature 
the same result is seen, for no marine animal except the cuttle¬ 
fish uses this principle of propulsion. Even the cuttle-fish cannot 
depend upon his jet to escape from his enemies, but for this relies 
upon his supply of ink with which he darkens the water about 
him. 

Prob. 186 . Compute the velocity and efficiency of a jet propeller driven 
by a 1-inch nozzle under a pressure of 150 pounds per square inch when A = 
1000 square feet and/ = 0.004. Compute also the efficiency when the diameter 
of the nozzle is 3 inches. 


Art. 187 . Paddle Wheels 

The method of propulsion by rowing and paddling is well 
known to all. The power is furnished by muscular energy within 
the boat, the water is the fulcrum upon which the blade of the 
oar acts, and the force of reaction thus produced is transmitted 
to the boat and urges it forward. If water were an unyielding 
substance, the theoretic efficiency of the oar should be unity, or, 




494 


Chap. 15. Naval Hydromechanics 


as in any lever, the work done by the force at the rowlock should 
equal the work performed by the motive force exerted by the 
man on the handle of the oar. But as the water is yielding, 
some of it is driven backward by the blade of the oar, and thus 
energy is lost. 

The paddle or side wheel so extensively used in river naviga¬ 
tion is similar in principle to the oar. The power is furnished by 
motor within the boat, the blades or vanes of the wheel tend to 
drive the water backward, and the reaction thus produced urges 
the boat forward. On first though it might be supposed that 
the efficiency of the method would be governed by laws similar 
to those of the undershot wheel, and such would be the case if 
the vessel were stationary and the wheel were used as an apparatus 
for moving the water. In fact, however, the theoretic efficiency 
of the paddle wheel on a boat is much higher than that of the 
undershot motor. 

The work exerted by the steam-engine upon the paddle wheels 
may be represented by PV, in which P is the pressure produced 
by the vanes upon the water, and V is their velocity of revolution ; 
and the work actually imparted to the boat may be represented 
by Pv, in which v is its velocity with respect to the water. Ac¬ 
cordingly the efficiency of the paddle wheel, neglecting losses 
due to foam and waves, is 


V v + vi 

in which v\ is the difference V — v, or the so-called “slip.” If 
the slip be o, the velocities V and v are equal, and the theoretic 
efficiency of the wheel is unity. The value of V is determined 
from the radius r of the wheel and its number of revolutions 
per second; thus V = 2-jrrn. 

On account of the lack of experimental data it is difficult to give 
information regarding the practical efficiency of paddle wheels con¬ 
sidered from a hydromechanic point of view. Owing to the water 
which is lifted by the blades, and to the foam and waves produced, 
much energy is lost. They are, however, very advantageous on ac¬ 
count of the readiness with which the boat can be stopped and re- 



The Screw Propeller. Art. 188 


495 


versed. When the wheels are driven by separate engines, as is some¬ 
times done on river boats, perfect control is secured, as they can be 
revolved in opposite directions when desired. Paddle wheels with 
feathering blades are more efficient than those with fixed radial ones, 
but practically they are found to be cumbersome, and liable to get 
out of order. In ocean navigation the screw has now almost entirely 
replaced the paddle wheel on account of its higher efficiency. 

Prob. 187 . The radius of the blades of a paddle wheel is 10.5 feet and the 
number of revolutions per minute is 24. If the efficiency is 75 percent, what 
is the velocity of the boat in miles per hour ? Show that for this case the 
slip is 33 percent of the velocity of the boat. 

Art. 188 . The Screw Propeller 

The screw propeller consists of several helicoidal blades 
attached at the stern of a vessel to the end of a horizontal shaft 
which is made to revolve by steam power. The dynamic pressure 
of the reaction developed between the water and the helicoidal 
surface drives the vessel forward, the theoretic work of the screw 
being the product of this pressure by the distance traversed. 
The pitch of the screw is the distance, parallel to the shaft, be¬ 
tween any point on a helix and the corresponding point on the 
same helix after one turn around the axis, and the pitch may be 
constant at all distances from the axis, or it may be variable. 
If the water were unyielding, the vessel would advance a distance 
equal to the pitch at each revolution of the shaft; actually, the 
advance is less than the pitch, the difference being called the 
“slip.” The effect thus is that the pressure P existing between 
the helical surfaces and the water moves the vessel with the 
velocity v, while the theoretic velocity which should occur is V, 
being the pitch of the screw multiplied by the number of revolu¬ 
tions per second. The work expended is hence PV or P(v + vy), 
if vy be the slip per second, and the work utilized is Pv. Ac¬ 
cordingly the efficiency of screw propulsion is, approximately, 

v 

e= - 

V + Vy 

which is the same expression as before found for the paddle 
wheel. Here, as in the last article, all the pressure exerted by the 



496 


Chap. 15. Naval Hydromechanics 


blades upon the water is supposed to act backward in a direc¬ 
tion parallel to the shaft of the screw, and the above conclusion 
is approximate because this is actually not the case, and also 
because the action of friction has not been considered. The 
practical advantage of the screw over the paddle wheel has been 
found to be very great, and this is probably due to the circum¬ 
stance that less energy is wasted in lifting the water and in form¬ 
ing waves. 

The pressure P which is exerted by the helicoidal blades upon 
the water is the same as the thrust or stress in the shaft, and the 
value of this may be approximately ascertained by regarding it 
as due to the reaction of a stream of water of cross-section a 
and velocity v, or p = wa („ + Pl )„/ g 

Another expression for this may be found from the indicated 
work k of the steam cylinders of the engines; thus 

P = k/v 

Numerical values computed from these two expressions do not, 
however, agree well, the latter giving in general a much less value 
than the former. 

In Art. 185 the work to be performed in propelling a vessel of 
fair form having the submerged surface A was found to be 

k = fAv 3 

If the value of v is taken from this equation and inserted in the 
expression for efficiency, there obtains 

i 

e =- - 

i+Vi (Af/k) 3 

which shows that e increases as V\, /, and A decrease, and as k 
increases. Or for given values of / and A the efficiency decreases 
with the speed. 

It has been observed in a few instances that the “slip” is nega¬ 
tive, or that V, as computed from the number of revolutions and pitch 
of the screw, is less than v. This is probably due to the circumstance 
that the water around the stern is following the vessel with a velocity 

so that the real slip is V — v + v' instead of V — v. The exist¬ 
ence of negative slip is usually regarded as evidence of poor design. 



Stability of a Ship. Art. 139 


497 


Twin screws are frequently used, and since these revolve in op¬ 



posite directions, the vessel can be more readily controlled. Fig. 188 
shows the position 
of the twin screws 
with respect to the 
rudder. On some 
of the recent high- 
powered turbine- 
driven steamships 
two and three 
screws all mounted 
on a single shaft 

have been em- Fi , r 188 

ployed. Two sets 

of engines, and two shafts, one on each side of the rudder, are 
often employed as in Fig. 188 , but a different arrangement of the 
shafts with respect to the hull of the ship permits the screws to 
be placed at considerable distances apart on the shafts, thus obtain¬ 
ing a greater efficiency than in the case of the single screw. 


Prob. 188 . A steamer having a submerged surface of 30 000 square feet 
is*propelled at 18 knots per hour by an expenditure of 6000 horse-powers. If 
the pitch of the screw is 20 feet, its number of revolutions 120 per minute, 
and / = 0.004, compute the number of lost horse-powers. 


Art. 189. Stability of a Ship 

In Art. 14 the general principles regarding the stability of a 
floating body were stated, and these are of great importance in 
the design of ships. The center of gravity is, of course, always 
above the center of buoyancy, and the metacenter must be above 
the center of gravity in order to insure stability. The distance 
between the metacenter and the center of gravity is denoted by 
m, and if the body be inclined slightly to the vertical at the angle 
#, the moment of the couple formed by the weight W of the body 
which acts downward through the center of gravity and the up¬ 
ward pressure W of the displaced water which acts through the 
center of buoyancy is Wm tan#. Hence m tan# is a measure 
of the stability of the body, and the greater its value, the greater 
is the tendency of the body to return to the upright position. 



















498 


Chap. 15. Naval Hydromechanics 


The metacentric height m cannot, however, be made very 
great, for the rapidity of rolling increases with it. When a 
floating body or ship is displaced from its vertical position, it 
rolls to and fro with isochronous oscillations like those of a pendu¬ 
lum, and the time of one oscillation from port to starboard is 

given by the formula _ 

/ = 7r V r 2 /mg 

in which r is the radius of gyration of the weight of the ship about 
a horizontal longitudinal axis passing through its center of 
gravity. Hence if m is large t is small and the ship rolls quickly; 



but if m is small t is large and the ship rolls slowly. The meta¬ 
centric height m for ocean vessels usually ranges from 2 to 15 
feet, about 6 or 8 feet being the usual value. 

The determination of the values of m and r for a ship is a labo¬ 
rious process, owing to its curved shape and the irregular distribution 
of its weight and cargo. The process will here be applied to the simple 
case of a rectangular prism of uniform density. Let h be the height 
and b the breadth of the prism, and / its length perpendicular to the 
plane of the drawing in Fig. 189 a. When the prism is in the vertical 
position, its depth of flotation is sh, if 5 is its specific gravity (Art. 13 ), 
and this is also the length of the immersed portion of the axis AB 
when the prism is inclined to the vertical at the angle 0, as in Fig. 
189 &. In the latter position the center of buoyancy D , being the 
center of gravity of the displaced water, is easily located, and 

^ _ b 2 tan0 _ sh b 2 tan 2 # 

12 sh ' 2 24 sh 

are its coordinates with respect to B, x being measured normal and y 
















Stability of a Ship. Art. 189 


499 


parallel to AB. The distance m from the center of gravity g to the 
metacenter M is then found to be 

m = -^-7(1 +1 tan 2 #) — 1 — s) 

If m is positive, the metacenter is above the center of gravity and the 
equilibrium is stable, for the moment Wm tan# restores the prism to 
the vertical position; if m is zero, the equilibrium is indifferent; if m 
is negative, the equilibrium is unstable, and the prism falls over. 

The square of the radius of gyration of the prism with respect 
to a horizontal longitudinal axis through G is its polar moment of 
inertia T y/(## 3 H- 7 ?# 3 ) divided by its volume Ibd, whence r 2 = yy(A 2 +# 2 ). 
For example, if h is 5 feet, b is 8 feet, and s is 0.5, the value of r 2 
is 7.42 feet 2 . The value of m to be used in the above formula for the 
time of one roll is that obtained by making 6 equal to zero, since that 
formula is strictly true only for small deviations from the vertical. 
For the above data this value of m is +0.88 feet, the plus sign denoting 
stability, and hence the time of one oscillation from port to starboard 
is t = 1.61 seconds. It is seen that t can be increased either by in¬ 
creasing r 2 or by decreasing m ; since a decrease in m is unfavorable 
to stability, it is usually preferable to increase r 2 . For instance, in 
loading a ship the cargo may be placed along the sides rather than near 
the middle of the hold, and this will increase r 2 , as the width of a ship 
is always greater than its depth. The general rule to promote sta¬ 
bility and prevent quick rolling is hence to place the cargo as far as 
possible from the center of gravity. 

The above formula for m shows that the moment Wm tan# which 
restores the floating prism to the vertical increases with the angle # 
up to a maximum value, then decreases, and when D arrives vertically 
beneath G, it becomes zero and the prism upsets. For the case where 
h = 5 feet, b = 8 feet, and s = 0.5, the value of m tan# is 0.00 feet 
for # = o°, 0.16 feet for # = io°, 0.37 feet for # = 20°, and 0.72 
feet for 0 — 30°; at # = 32 0 the corner of the prism becomes immersed 
so that the formula no longer holds, but up to this point the moment 
constantly increases. From the above expression for m the solution 
of Prob. 14 is readily made. 

Prob. 189 #. An open rectangular wooden box caisson of length /, breadth 
b, and depth d has sides of mean thickness #1 and a bottom of thickness d x . 
Deduce formulas for the metacentric height m and the squared radius of 
gyration r 2 . Compute m, r 2 , and t for a numerical case. 



500 


Chap. 15. Naval Hydromechanics 


Art. 190. Action of the Rudder 



The action of the rudder in steering a vessel involves a prin¬ 
ciple that deserves discussion. In Fig. 190 is shown a plan of 

a boat with the rudder turned to the 
starboard side, at an angle 6 with the 
line of the keel. The velocity of 
the vessel being v, the action of the 
water upon the rudder is the same as 
if the vessel were at rest and the water 
in motion with the velocity v. Let W 
be the weight of water which produces 
dynamic pressure against the rudder, due to the impulse W ' v/g 
(Art. 152 ). The component of this pressure normal to the rud- 

der 1S P = Wv sin 0 /g 

and its effect in turning the vessel about the center of gravity 

C is measured by its moment with reference to that point. Let 

b be the breadth of the rudder and d the distance CH between the 

center of gravity and the hinge of the rudder, then the lever arm 

of the force P is 717,7 /1 

l "2 b I cl cose/ 

and accordingly the turning moment is 

M = \W(b sin6 -\-d sin20 )v/g 

To determine that value of 6 which produces the greatest effect 
in turning the boat the derivative of M with respect to 6 must 
vanish, which gives 

cosd = — — + 

Sd 

and from this the value of 6 is found to be approximately 45 0 , 
since d is always much larger than b. 

Values of the angle 6 for several values of the ratio b/d may now 
be computed as follows: 



b/d= l i 

cos 0=0.6825 0.6916 

0 = 46° 58' 46° 15' 


_1 
1 0 


0.6947 
46° 00' 


_1 
1 OTT 

O.7069 
45 ° °i' 


0.7071 

45 ° 







Tides and Waves. Art. 191 


501 


which shows that about 45 0 is the advantageous angle. In practice 
it is usual to arrange the mechanism of the rudder so that it can only 
be turned to an angle of about 42 0 with the keel, for it is found that the 
power required to turn it the additional 3 0 or 4 0 is not sufficiently com¬ 
pensated by the slightly greater moment that would be produced. 
The reasoning also shows that intensity of the turning moment in¬ 
creases with v, so that the rudder acts most promptly when the boat 
is moving rapidly. For the same reason a rudder on a steamer pro¬ 
pelled by a screw is not required to be so broad as one on a boat driven 
by paddle wheels, for the effect of the screw is to increase the velocity 
of the impinging water, and hence also to increase the dynamic pres¬ 
sure against the rudder. 

Prob. 190 . Explain how it is that a boat can sail against the wind. What 
is the influence of the keel in this motion ? 


Art. 191 . Tides and Waves 

The complete discussion of the subject of waves might, like 
many other branches of hydraulics, be expanded so as to em¬ 
brace an entire treatise, while there can be here given only 
the briefest outline of a few of the most important principles. 
There are two classes or kinds of waves, the first including the 
tidal waves and those produced by earthquakes or other sudden 
disturbances, and the second those due to the wind. The daily 
tidal wave generated by the attraction of the moon and sun orig¬ 
inates in the South Pacific Ocean, whence it travels in all direc¬ 
tions with a velocity dependent upon the depth of water and the 
configuration of the continents, and which in some regions is as 
high as 1000 miles per hour. Striking against the coasts, the tidal 
waves cause currents in inlets and harbors, and if the circum¬ 
stances were such that their motion could become uniform and 
permanent, these might be governed by the same laws which 
apply to the flow of water in channels. Such, however, is rarely 
the case; and accordingly the subject of tidal currents is one 
of much complexity and not capable of general formulation. 

The velocity of a tidal wave on the ocean is VgD, where D 
is the depth of the water. When such a^wave rolls over the 
land, the greatest velocity it can have is Vgff, where d is its depth, 



502 


Chap. 15. Naval Hydromechanics 


this being the case of the bore (Art. 139 ). The velocity of a 
wave which is produced by a sudden disturbance in a channel of 
uniform width has also been found to be VgZ), where D is the 
depth of the water. 

Rolling waves produced by the wind travel with a velocity which 
is small compared with those above noted, although in water where 
the disturbance can extend to the bottom, it is generally supposed that 
their velocity is VgH. Upon the ocean the maximum length of such 
waves is estimated at 550 feet and their velocity at about 53 feet per 
second. For this class of waves it is found by observation that each 
particle of water upon the surface moves in an elliptic or circular orbit, 
whose time of revolution is the same as the time of one wave length. 



Thus the particles on the crest of a wave are moving forward in the 
direction of the motion of the wave, while those in the trough are mov¬ 
ing backward. When such waves advance into shallow water, their 
length and speed decrease, but the time of revolution of the parti¬ 
cles in their orbits remains unaltered, and as a consequence the slopes 
become steeper and the height greater, until finally the front slope be¬ 
comes vertical and the wave breaks with roar and foam. Below the 
surface the particles revolve also in elliptic orbits, which grow smaller 
in size toward the bottom. The curve formed by the vertical sec¬ 
tion of the surface of a wave at right angles to its length is of a cycloidal 
nature. 

The force exerted by ocean waves when breaking against sea 
walls is very great, as already mentioned in Art. 155 , and often proves 
destructive. If walls can be built so that the waves are reflected with¬ 
out breaking, as is sometimes possible in deep water, their action is 
rendered less injurious. Upon the ocean waves move in the same di¬ 
rection as the wind, but along shore it is observed that they generally 
move normally toward it, whatever may be the direction in which the 
wind is blowing. The force of wave action is felt at depths of over 
100 feet below the surface, for sand has been brought up from depths 





Tides and Waves. Art. 191 


503 


of So feet and dropped upon the decks of vessels. Shoals also cause 
a marked increase in the height of waves, even when such shoals are 
500 feet or more below the water surface. 

Prob. 191 a. I11 a channel 6.5 feet wide, and of a depth decreasing 1.5 feet 
per 1000 feet, Bazin generated a wave by suddenly admitting water at the 
upper end. At points where the depths were 2.16, 1.85, 1.46, and 0.80 feet, 
the velocities were observed to be 8.70, 8.67, 7.80, and 6.69 feet per second. 
Do these velocities agree with the theoretic law? 

Prob. 1916 . Show that the values of / given in Art. 175 for use in the 
formula F—fv 2 are to be multiplied by 5.255 when v is in meters per second 
and F in kilograms per square meter. 

Prob. 191 c. Compute the metric horse-power required for a velocity of 
25 kilometers per hour for a boat which has a submerged area of 237 square 
meters. 

Prob. 191 ^. A ship rolls from starboard to port in 7.5 seconds. If the 
metacentric height m is 2.4 meters, what is the value of the transverse radius 
of gyration of the ship ? How much must the radius of gyration be increased 
in order to increase the time of rolling 15 percent ? 


504 


Chap. 16. Pumps and Pumping 


CHAPTER 16 
PUMPS AND PUMPING 

Art. 192. General Notes and Principles 

Among the simple devices for raising water that have been 
used for many centuries, and which may be called lift pumps 
in a general way, are the sweep and windlass, buckets attached 
to a revolving wheel, the chain and bucket pump where the 
buckets move in a cylinder, and the Archimedian screw. The 
chain and bucket pump was probably first used by the Chinese 
in the form of an inclined trough in which moved the buckets 
attached to the endless chain, and this device in various forms 
has been used in all countries for lifting water from wells. The 
Archimedian screw, invented by the great engineer Archimedes 
when he was in Egypt, about 240 b.c., consists of a tube wound 
spirally around an inclined cylinder. When the lower end is 
placed under water and the cylinder revolved, the water is lifted 
and flows out of the upper end of the tube. This screw pump 
is still in use in northern Egypt, and it is said to be a satisfactory 
apparatus for a low lift. 

The fact that water would sometimes rise into a space from 
which the air had been removed was known at a remote antiquity, 
and this was frequently explained by the statement that “ nature 
abhors a vacuum.'' It was not until the middle of the seventeenth 
century that the true reason of this phenomenon was explained 
through the researches of Torricelli and Pascal (Art. 4 ), but 
prior to this time a rude form of suction pump, made by attach¬ 
ing a pipe to a bellows at the opening where the air usually enters, 
was used in both France and Germany. In 1732 the first true 
suction and lift pump was devised by Boulogne, and a little later 
the suction and force pump came into use. 


General Notes and Principles. Art. 192 505 

The force pump is a device for raising water by means of 
pressure exerted on it by a piston. The syringe, which has been 
known from very early times, is an example of this principle, 
but the first true force pump was invented in Egypt about 250 
B.C., by Ctesibius, a Greek hydraulician, and the description of 
it given by Vitruvius indicates that it was used to some extent 
by the Romans. The early force pumps were placed with their 
cylinders below the level of the water to be lifted, and had valves 
which closed under the back pressure of the water. By placing 
the cylinders above the water level and utilizing the principle of 
suction, the suction and force pump originated. 

All devices for raising water may be classified under the three 
principles above mentioned: that of lifting in buckets, drawing 
it up by suction, or forcing it up by pressure, or under combina¬ 
tions of these. The lift or bucket principle is mainly employed 
for small quantities of water and has only a limited use in en¬ 
gineering practice. The suction principle, combined with lift 
or pressure, is extensively used, but in no event can the height 
of the suction exceed 34 feet, for it is the atmospheric pressure that 
causes the water to rise when the air above it is exhausted ; under 
this principle also may be put injector pumps which operate under 
the action of negative pressure-head (Art. 31 ). The principle of 
direct pressure governs not only the force pump, but rotary and 
centrifugal pumps and also the devices for raising water by com¬ 
pressed air. 

Whenever water is raised from a lower to a higher level, an 
amount of work must be expended greater than the theoretic 
work required to lift the given weight of water through the given 
height. The excess, called the lost work, is spent in overcoming 
resistances of friction and inertia. In designing pumps it is the 
object to reduce these losses to a minimum, so that the greatest 
economy in operation may result. The subject will here be 
mainly considered from a hydraulic standpoint, the object being 
to set forth the fundamental principles by which hydraulic losses 
may be avoided as far as possible. 

Let W be the weight of water raised per second and h the 


506 


Chap. 16. Pumps and Pumping 


height of the lift, then the useful work per second k is IF//. Let 
the total work expended per second be called K , then the efficiency 
of the apparatus is e = k/K. The work K to be considered here 
is that delivered to the pump and does not include that lost in 
transmission from the motor, since this, of course, is not fairly 
chargeable against the pump or lifting apparatus. If K be re¬ 
placed by Will + //'), where h f is the head lost in overcoming the 
frictional resistances, then the efficiency may be written 

_ k _ h 
6 K h + ti 

which is less than unity, since //' cannot be made zero. 

The power required to operate a pump to raise the weight W 
of water per second through the height h is easily computed if 
the efficiency of the pump is known. For example, to raise 150 
gallons per second through a height of 20 feet with a pump having 
an efficiency of 62 percent, the work which must be imparted to 
the pump per second is 

K = k/e = (150 X 8.335 X 20)/0.62 = 40 340 foot-pounds, 
and this, divided by 550, gives 73.3 horse-powers. 

Prob. 192 . A pump raises 20.5 cubic feet of water per second through 
a height of 127.5 feet. The lost head in the pump and pipes amounts to 
13.5 feet. Compute the efficiency of the pumping plant and the power re¬ 
quired to operate it. 



Art. 193 . Raising Water by Suction 

The term “ suction ” is a misleading one unless it be clearly 
kept in mind that water will not rise in a vacuum tube unless 
the atmospheric pressure can act underneath it. For example, 
no amount of rarefaction above the surface of the water in a 
glass bottle will cause that water to rise. When the tube is 
inserted into a river or pond, however, the water will rise in it 
when a partial vacuum is formed, since the atmospheric pressure 
which is transmitted through the water pushes it up until equilib¬ 
rium is secured (Art. 4 ). The mean atmospheric pressure of 14.7 
pounds per square inch at the sea level is equivalent to a height 



Raising Water by Suction. Art. 193 


507 


of water of 34 feet, and this is the limit of raising water by suc¬ 
tion alone. In practice this height cannot be reached on account 
of the impossibility of producing a perfect vacuum, and it is found 
that about 28 feet is the maximum height of suction lift. 

The height of the water barometer varies with the state of 
the weather, with the elevation above sea level, and with the 
temperature. The value of 34 feet is that corresponding to a 
reading of 30 inches on the mercury barometer at a temperature 
of 32 0 Fahrenheit. For higher temperatures more or less vapor 
is evaporated from the water surface and fills the suction tube, 
so that a complete vacuum cannot be formed. When the mercury 
barometer reads 30 inches, the water barometer is only 33.4 feet 
if the temperature of the water is 6o° Fahrenheit, 32.4 feet at 
90°, about 30 feet for 120°, about 23 feet for 160°, about 6 feet 
for 200°, and for 212 0 its height is zero, since the tube is then 
filled with steam. Hence water at the boiling-point cannot be 
raised by suction. 

Fig. 193 gives two diagrams illustrating the principle of action 
of the common suction and lift pump. It consists of two verti¬ 
cal tubes BD and BE , the 
former being called the suc¬ 
tion pipe and the latter the 
pump cylinder. The piston 
A in the pump cylinder has 
a valve opening upward, and 
the valve B at the top of the 
suction pipe also opens up¬ 
ward. In the left-hand dia¬ 
gram the piston is descending, 
the valve A being open and 
B being closed under the pres¬ 
sure of the air in the space 
between them. In the right- 
hand diagram the piston is 
ascending, the valve A being closed by the pressure of the air or 
water above it, while B is open, owing to the excess of atmos- 



Fig. 193. 

















































508 


Chap. 16. Pumps and Pumping 


pheric pressure in BD above that in AB. In the first diagram 
the piston has made only one or two strokes, so that the water 
has risen but a short distance in the suction pipe. In the 
second diagram the piston has made a sufficient number of 
strokes so that the pump cylinder is full of water which is flowing 
out at the spout E. 

Let h\ be the distance from the water level D to the lowest 
position of the piston; this is called the height of lift by suction. 
Let h 2 be the height from the lowest position of the piston to the 
spout where the water flows out; this is called the height of lift 
by the piston. The distance hi + h 2 is the vertical height through 
which the water is raised, and if W be the weight of water raised 
in one second, the useful work per second is W(h\ -f- h 2 ). The 
energy imparted to the pump through the piston rod is always 
greater than this useful work, since energy is required to overcome 
the frictional resistances due to the motion of the water and pis¬ 
ton, as also to overcome the resistance of inertia in putting them 
into motion. 

To discuss the action of the pump in detail, let l be the stroke 
of the piston, that is, the distance between its highest and lowest 
positions. Let. A be the area of the cross-section of the pump 
cylinder and a that of the suction pipe. Let the piston be sup¬ 
posed to be at its lowest position at the beginning of the operation 
when no water has been raised in the suction pipe above the level 
D in Fig. 193 . On raising the piston through the stroke l it 
describes the volume Al, and the volume of air alh now has the 
volume Al + a(hi — x ) in which x is the height through which 
the water rises during the upward stroke. Let h a be the height 
of a water barometer corresponding to the air pressure above the 
water level at the beginning of the stroke, then h a — y is the pres¬ 
sure-head at the end of the stroke. Since, by Mariotte’s law, 
the pressure of a given quantity of air is inversely as its volume, 
(h a — x)/h a equals ahi/(Al -j- alii — ax ), whence, 

x 2 — (rl + hi + h^x + rlh a = o 

in which r represents the ratio A/a. For example, let A be 
8 and a be 2 square inches, or r = 4, let hi be 20 and / be 1.5 feet; 


Raising Water by Suction. Art. 193 


509 


then for h a = 34 feet, the water rises during the first upward 
stroke to the height x = 3.6 feet. For the second upward stroke 
h a is 34.0 — 3.6 = 30.4 feet and h\ is 20.0 — 3.6 = 16.4 feet; 
then the formula gives x = 3.7 feet, so that the water level now 
stands 7.3 feet above its original level D. Proceeding in like 
manner, it is found that at the end of the third upward stroke 
the water stands at 11.2 feet above its original level. Similarly at 
the end of the fourth upward stroke it is found to be 15.3 feet 
above D, while at the end of the fifth upward stroke it has reached 
a height of 19.8 feet above its original level. During the progress 
of the sixth upward stroke the water enters the pump cylinder, 
during the next downward stroke it flows through the piston 
valve, and in the seventh upward stroke the water above the 
piston is lifted and flows out through the spout. 

The preceding discussion supposes that there is no leakage of 
air through and around the piston, but this cannot be attained in 
practice; hence the degree of rarefaction below the piston is never so 
great as the above formula gives, and the number of strokes required 
to elevate the water above the valve B is larger than the computed 
number. When the suction height is greater than 25 feet, it becomes 
difficult to secure sufficient rarefaction to lift the water, and hence a 
foot valve, also opening upward, is placed in the suction pipe below 
the water level D. The pump cylinder and suction pipe can then be 
primed, or filled with water from above, and after this is done there 
will be no difficulty in operating the pump. If there is no foot valve, 
it will be necessary to have a very long piston stroke in order to start 
the pump, but with a foot valve the stroke of the piston may be 
any convenient length. 

The action of this pump is intermittent, and water flows from the 
spout only during the upward stroke of the piston. When there are 
N upward strokes per minute, the discharge in one minute is NAl, 
if the piston and its valve be tight. The useful work per minute is 
NwAlQ^A-h^), if w be the weight of a cubic unit of water. When l 
and h l A~h 2 are in feet, A in square feet, and w in pounds per cubic foot, 
the horse-power expended in this useful work is 

IIP = NwAlQi 1+ lh)/33 000 

and to this must be added the horse-power required to overcome 
the resistances of friction and inertia. This additional power often 


510 


Chap. 16. Pumps and Pumping 


amounts to as much as that needed for the useful work, and in this case 
the efficiency of the pump is 50 percent. Suction and lift pumps 
are of numerous styles and sizes, the simplest being of square wooden 
tubes or of round tin-plate tubes with leather valves, and these can 
be readily made by a carpenter or tinsmith. They are mainly used 
for small quantities of water and for temporary purposes. 

Prob. 193 . The diameter of the pump cylinder is 8 inches and that of the 
suction pipe is 6 inches, while the vertical distance from the water level to the 
spout is 23 feet. If the pump piston makes 30 upward strokes per minute, 
each 9 inches long, what horse-power is required to operate the pump if its 
efficiency is 45 percent ? 

Art. 194 . The Force Pump 

A force pump is one that has a solid piston which can trans¬ 
mit to the water the pressure exerted by the piston rod and thus 
cause it to rise in a pipe. The early force pumps had little or no 
suction lift, as the pump cylinder was immersed in the body of 

water which furnished the sup¬ 
ply, but the modern forms 
usually operate both by suction 
and pressure, the former occur¬ 
ring in a suction pipe and the 
latter in the pump cylinder. 
Fig. 194(7 shows the principle of 
action of the common vertical 
single-acting suction and force 
pump in which there is no water 
above the piston. In the left- 
hand diagram the piston is as¬ 
cending, and the water is rising 
in the suction pipe BD under 
the upward atmospheric pres¬ 
sure ; this ascent of the water 
occurs in exactly the same manner as explained in Art. 193 , and 
after several strokes its level rises above the suction valve B. 
The right-hand diagram shows the piston descending and forcing 
the water up the discharge pipe CE. At C, where this pipe 




































































The Force Pump, Art. 194 


511 


joins the pump cylinder, is a check valve which closes on the 
upward stroke and thus prevents the water in CE from returning 
into the pump cylinder, while it opens on the downward stroke 
under the upward pressure of the water. 


Let A be the area of the cross-section of the pump cylinder 
and l the length of the stroke of the piston. Then at each upward 
stroke a volume of water equal to Al is raised through the suction 
pipe, and in each downward stroke the same volume is raised in 
the discharge pipe. If h be the total lift above the water level 
D and w the weight of a cubic unit of water, the work done in each 
double stroke is wAlh. If there be made N double strokes per 
minute, the useful work per minute is NwAlh. When all dimen¬ 
sions are in feet, the horse-power required to do this useful work 
is found by dividing this quantity by 33 000, and the actual 
horse-power required to run the pump is greater than this by the 
amount needed to overcome the frictional resistances. This 
additional power will depend upon the length of the suction and 
discharge pipes, the speed at which the pump is operated, the 
friction along the sides of the piston, the losses of head in the 
passage of the water through the valve openings, and the losses 
of energy due to putting the water into motion at each stroke. 
The efficiency of single-acting suction and lift pumps hence varies 
between wide limits, 90 percent or more being obtained only for 
very low speeds and lifts, while for high speeds and lifts it may 
be 20 percent or less. 


E 




fC} 


B 

[D 

Fig. 1946. 










* 



Fig. 194c. 


The cylinder of the single-acting pump may be placed hori¬ 
zontal, as seen in Fig. 1945 , where BD is the suction pipe and 



























































512 


Chap. 16. Pumps and Pumping 


CE the discharge pipe. When the piston moves toward the left, 
the suction valve B opens and the check valve C closes; when it 
moves toward the right, B closes and C opens. The discharge 
is intermittent, as in the previous case, but the horizontal position 
of the piston sometimes renders the connection of the piston rod 
to the motor more convenient. If the height of the suction lift be 
equal to that of the discharge lift, the force required to move the 
piston will be the same in each stroke and the pump will work 
with less shock than where the two lifts are unequal. Usually, 
however, the height of the discharge lift is greater than that of 
the suction lift, and the force required to move the piston is then 
the greatest when it moves from left to right in Fig. 1945 . In 
order to equalize the forces exerted by the motor the duplex pump 
was devised; this consists of two single-acting cylinders placed 
side by side and connected to the same suction and discharge 
pipe, the pistons moving so that one exerts suction while the 
other is forcing the water upward. Three single-acting cylinders 
are also sometimes connected with the same suction and dis¬ 
charge pipe, in which case it is called the triplex pump. Duplex 
and triplex pumps give a more nearly continuous flow of water 
in both the suction and discharge pipes, and thus diminish the 
shocks that occur in a pump with one cylinder, while the efficiency 
is materially increased because the losses due to starting and 
stopping the columns of water are in large part avoided. 

A double-acting pump is one having a single cylinder in which 
a solid piston or plunger exerts suction and pressure in both 
strokes and thus gives a nearly continuous flow through suction 
and discharge pipes. Fig. 194 d shows the form known as the 
piston pump, while Fig. 194 c is that called the plunger pump, 
the piston being replaced by a long cylinder moving in a short 
stuffing box AA. In both figures D is the suction pipe and E 
the discharge pipe. When the piston moves from left to right, 
the valves Bi and C2 open, while Bo and Ci close; when it moves 
in the opposite direction, B 2 and Ci open, while By and C 2 close. 
The plunger pump was invented in the seventeenth century, and 
its advantages over the piston type are so great that it is now 


The Force Pump. Art. 194 


513 


extensively used for large pumping machinery. The cylinder of 
the piston pump must be bored to an exact and uniform size, and 
its piston must be carefully packed, while in the plunger pump 
only the short length of the stuffing box is bored and packed, the 




plunger itself having no packing. The water lifted in one stroke 
of either pump is Al , where A is the area of the piston and / the 
length of its stroke, provided there is no leakage past the packing. 

For all these forms of pumps a foot valve should be placed in the 
suction pipe, if the suction lift exceeds 20 feet, in order that the pump 
may be readily primed (Art. 193 ). To reduce the shocks that occur 
to a certain extent even in the double-acting pumps, an air chamber 
is frequently attached to the discharge pipe so that the confined air 
may distribute and lessen the shock that would otherwise be concen¬ 
trated on the end of the discharge pipe. Fig. 194 c shows such an air 
chamber attached to a single-acting pump; in the upper part of it 
is seen the compressed air which is receiving the pressure from the 
piston. After the check valve C closes the pressure of this air main¬ 
tains the flow up the discharge pipe E, and hence the air chamber 
helps to avoid the losses due to intermittent flow. A duplex pump 
or a double-acting pump, when provided with an air chamber of proper 
size, will work very smoothly. 

Prob. 194 . Consult Ewbanks’ Hydraulics and Mechanics (New York, 
1847), and describe a method of raising water through a low lift by means of 
a frictionless plunger pump. Ewbank notes that a stout young man weigh¬ 
ing 134 pounds raised cubic feet per minute with this machine to a height 
of ii| feet, and worked at this rate nine hours per day. If the efficiency of 
this pump was unity, what horse-power did the stout young man exert ? Was 
his performance high or low? 



























514 


Chap. 16. Pumps and Pumping 


Art. 195 . Losses in the Force Pump 


A reliable numerical computation of the hydraulic losses 
of energy in the force pump cannot be made without knowing 
the constants to use in finding the losses of head due to the 
valves (Art. 92 ), and these have been experimentally determined 
for only a few special forms. The valves shown in most of the 
figures of the preceding articles are simple flap valves, but poppet 
valves are more generally used, and Fig. 194 c indicates such. In 
passing through a valve the water loses energy in friction, and also 
in impact due to the subsequent expansion. Since pumps are 
made in numerous forms having different details, general discus¬ 
sions of losses are difficult to make. The attempt will, however, 
be undertaken for the plunger force pump of Fig. 194 c. Let h 
be the total height through which the water is lifted by both 
suction and pressure, and h' be the sum of all the hydraulic losses 
of head. Let K be the energy delivered per second to the piston 
rod, k' the energy expended in friction in the stuffing boxes of the 
piston rod and plunger, q the discharge per second, and w the 
weight of a cubic unit of water. Then 



K = k' + wq[h + ^- 

\ 2 g 


and the pump should be so arranged as to make the losses k' and 
h' as small as possible. Only the hydraulic losses will be con¬ 
sidered in the following discussion. 

By means of the principles of Chap. 7 a rough formulation 
of the elements that make up the lost head h f can be effected, 
supposing the flow in the pipes to be steady. Let h be the length, 
di the diameter, and Vi the velocity for the suction pipe, and / 2 , 
d 2 , and v 2 the same things for the discharge pipes. Let 2n be 
the number of valves in the suction and discharge chambers 
(Fig. 194 c), all being taken of the same size, and let V denote 
the velocity of the water through each valve opening. Let these 
chambers be so large that the velocity of the water through them 
is very small compared to that in the pipes and valve openings. 



(195) 






Losses in the Force Pump. Art. 195 


515 


gives all the hydraulic losses of head. In the first parenthesis 
m indicates the loss due to entrance at the foot of the suction pipe 
(Art. 89), fh/di the friction loss in the suction pipe (Art. 90), 
and i the loss due to expansion (Art. 76) as the water enters the 
suction chamber BiB 2 . In the second parenthesis m' indicates 
the loss due to the open valves (Art. 92) and i that due to sudden 
expansion as the water enters the pump cylinder through the 
suction valves and the discharge chamber CiC 2 through the dis¬ 
charge valves. The last term gives the loss due to friction in the 
discharge pipe. If there is an air chamber on the discharge pipe, 
another term might be introduced, but as the effect of the air 
chamber in reducing water hammer is a beneficial one, this term 
need not be used. The starting and stopping of the piston brings 
in other losses of energy, but as these are not hydraulic losses 
they will not be considered here. 

When the pipes are long, the losses due to pipe friction will 
far exceed those in the pump, and are not fairly chargeable against 
it as a machine; hence in order to consider the pump alone 
the lengths h and / 2 may be made equal to zero, as also m in the 
first parenthesis. Then formula (195) becomes 

7 ! 2 V- 

^=^-+ 2 (w'+ I ) — 

2 g 2 g 


in which the first term of the second member gives the loss of 
head in entering the suction chamber, and the second those oc¬ 
curring in entering and leaving the pump cylinder. This equa¬ 
tion appears, at first thought, to indicate that a suction chamber 
is not a hydraulic advantage, although it is known to afford a 
practical advantage in causing the valves to operate successfully, 
as also in permitting ready access to them. If a be the area of 
each valve opening, and a\ that of the suction pipe, then afO\ must 
equal \naV , since the same quantity of water passes per second 
through the suction pipe and through \n valves. Accordingly the 
total loss of head in the pump may be written 





2 g 





516 


Chap. 16. Pumps and Pumping 


which clearly shows that this loss decreases as the number of 
valves increases, when a is kept constant. Therefore the suction 
and discharge chambers may be made to give a hydraulic advan¬ 
tage, either by using many valves of a given size or by making 
the total valve area na sufficiently large, since h’ is thus diminished. 
The number of valves will usually be 8, 12, or 16. 


As a numerical example, take a plunger force pump, like Fig 
194 c, having a piston area A = 0.84 square feet, and a stroke of 1.25 
feet, the number of single strokes per minute being 30. The volume 
of water lifted per second is hence 30 X 0.82 X 1.25/60 = 0.525 cubic 
feet. Let the diameter of the suction pipe be 10 inches and the area 
of its cross-section a l = 0.545 square feet. The mean velocity in the 
suction pipe is then 0.525/0.545 = 0.96 feet per second. Let there be 
12 valves in the suction chamber, so that n = 6, and let the area of 
each valve opening be a = 8 square inches = 0.0556 square feet. 
The velocity through each of the open valves is then V = 0.525/3 
X 0.0556 = 3.15 feet per second. As Art. 92 does not give the values 
of m! for poppet valves, it may be here noted that the experiments 
of Bach* indicate that they range from 1.1 to 2.8, depending upon the 
height of valve lift and the width of the seat. Taking 2 as a mean 
value of m', the lost head in the pump is 


ti = 0.01555 


i+8X3 ( °' 545 y 

L V6X 0.0556/ 


0.96 2 = 0.96 feet. 


The useful head h, when the lengths of the suction and discharge pipes 
are disregarded, is probably about 3 feet, so that the hydraulic effi¬ 
ciency is e = k/(h+ti) = 0.75. If the lengths of the vertical suction 
and discharge pipes be each 20 feet and their diameters be 10 inches, 
the useful head h is about 43 feet and from ( 195 ) the value of ti is 
found to be about one foot, so that the hydraulic efficiency is about 
0.97. The velocity-head z > 2 2 /?g which is lost at the top of the discharge 
pipe is here only 0.01 feet, so that it is unnecessary to consider it in 
determining the efficiency. 

This discussion shows that the losses of head in force pumps 
may be made very slight by running them at low speeds in order that 
the velocity may be small. It shows that the losses decrease as the 
areas of the valve openings and their number are increased. It shows 


* Zeitschrift deutscher Ingenieur Verein, 1886, p. 421. 





Pumping Engines. Art. 196 


517 


that, for vertical suction and discharge pipes, the efficiency increases 
with the useful lift //, if the velocity in the pipes is the same for 
different lifts. These conclusions are verified by experiments, some 
of which will be noted in the next article. Since the flow through the 
valves and pump cylinder is not quite steady, numerical computations 
like the above cannot, however, be expected to give more than rough 
approximate results; nevertheless such results are useful in indicating 
the influence of the resistances upon the efficiency. 

Prob. 195 . For the above numerical example, compute the horse-power 
required to run the pump when the useful lift is 43 feet, assuming that 3 per¬ 
cent of that power is expended in overcoming friction in the stuffing boxes. 

Art. 196 . Pumping Engines 

The steam engine was invented and perfected through the 
desire to devise methods of pumping water better than those in 
which the power of men and horses was used. Worcester in 
1633, and Papin in 1695, used the direct pressure of steam upon 
water in a cylinder, and Savery in 1700 used both such pressure 
and the partial vacuum caused by the condensation of the steam. 
Newcomen in 1705 used a piston, on one side of which steam 
was applied and condensed, the motion of the piston being com¬ 
municated by a walking beam to the piston rod of a pump. Watt, 
about 1775, introduced the crank, the parallel motion, the cut-off, 
the governor, and other improvements; he also. brought the 
steam to both sides of the piston, thus making the engine double- 
acting. The first important application of the steam engine 
was in operating pumps to drain mines, but it soon came into 
use in all branches of industry where power was needed. Its 
influence on modern progress has been great. 

The modern pumping engine consists of one or more steam 
cylinders connected to the same number of pump cylinders by 
piston rods, so that the steam pressure is directly transmitted 
through them to the water. It is important that the pressure 
in the water cylinder should be maintained nearly constant 
during the length of the stroke, and hence the steam should not be 
used expansively in the usual way; to insure constant steam 
pressure some form of compensator is used. The water cylinders 


518 


Chap. 16. Pumps and Pumping 


are usually of the plunger type, and these are connected to the 
same suction and discharge pipes, an air chamber being placed 
on the latter to relieve the pump chambers of shock and to in¬ 
sure steady flow. The boilers, steam cylinders, and water cyl¬ 
inders constitute one machine or apparatus called a pumping 
engine. The efficiency of this apparatus is low, for it is equal to 
the product of the efficiencies of its separate parts. The efficiency 
of the furnace and boiler is about 75 percent in the best designs, 
the efficiency of the steam cylinders about 30 percent, and that 
of the water cylinders about 80 percent, so that the efficiency of 
the pumping engine as a whole is only 18 percent. This means 
that only 18 percent of the energy of the fuel is utilized in lifting 
the water, and this figure is, indeed, a high one, for many pump¬ 
ing plants are operated with an efficiency of less than 10 percent. 

The term “duty” is often employed as a measure of the per¬ 
formance of a pumping engine, instead of expressing it by an 
efficiency percentage. This term was devised by Watt, who 
defined duty as the number of foot-pounds of useful work pro¬ 
duced by the consumption of 100 pounds of coal. On account 
of the variable quality of coal a more precise definition of duty 
was introduced in 1890 by a committee of the American Society of 
Mechanical Engineers, namely, that duty should be the number 
of foot-pounds of work produced by the expenditure of 1 000 000 
British thermal heat units. One British thermal heat unit is 
that amount of energy which will raise one pound of pure water one 
Fahrenheit degree in temperature when the water is at or near 
the temperature of maximum density (Art. 3 ); this amount of 
energy is 778 foot-pounds, and this constant is called the me¬ 
chanical equivalent of heat. The duty of a perfect pumping 
engine, in which no losses of any kind occur, would be 778 000 000 
foot-pounds. The highest duty obtained in a test is about 
180000000 foot-pounds, and the efficiency of such an engine is 
180/778 = 0.23.* Common pumping engines have duties ranging 
from 120000000 to 60000000, the corresponding efficiencies 
being from 15 to 7.5 percent. The modern definition of duty 

Transactions American Society of Civil Engineers, vol. 73, 1911. 


Pumping Engines. Art. 196 


519 


agrees with that of Watt, if the coal used be of such quality that 
one pound of it possesses a potential energy of io ooo British 
heat units, which is somewhat less than that obtainable from 
average coal. The higher the duty of a pumping engine the 
greater is the amount of work that can be performed by burning 
a given quantity of coal. A high-duty engine is hence econom¬ 
ical and a low-duty engine is wasteful in coal consumption, but 
the first cost of the former is much greater than that of the latter. 

A duty test of a pumping engine consists in determining the 
number of heat units furnished by a given quantity of coal, the 
quantity of water lifted by the pump, the leakage past the piston 
packing, the pressure-heads in the suction and discharge pipes, 
the indicated horse-power of the steam cylinders, and many other 
minor quantities needed for estimating the efficiency of the boiler 
and steam part of the apparatus. The usual method of deter¬ 
mining the discharge is by the displacement of the piston or 
plunger; if A be the area of its cross-section, l the length of the 
stroke, N the number of single strokes during the test, and T 
the number of seconds during which the test lasted, then NAl is 
the total quantity of water lifted, and 

q = cNAl/T 

is the quantity lifted per second, c being a coefficient which takes 
account of the leakage or slip past the plunger. The value of c 
is to be found by removing one of the cylinder heads and admit¬ 
ting water on the other side of the plunger, and its value is usually 
from 0.99 to 0.95 in new pumps, The total pressure-head II is 
found from H = (f h ±h l + d) 

where hi and A 2 are the pressure-heads corresponding to the mean 
readings of the gages on the suction and discharge pipes and d 
the vertical distance between the centers of the gages; here the 
plus sign is to be used when the corresponding pressure is below 
and the minus sign when it is above that of the atmosphere. 
The total work done by the pump during the trial is then cNAl ■ II 
and then the duty of the pumping engine 

Duty = 1 000 000 c.VA// 7 /heat units, 


520 


Chap. 16. Pumps and Pumping 


in which the denominator is determined by the thermodynamic 
tests made on the boiler and steam engine. The capacity of the 
pump, or the quantity of water lifted in 24 hours, is 24 X 3600 X q. 

The efficiency of pump cylinders, which are tested in the above 
manner, is usually found by dividing the work wqH done by them 
in one second by that done by the steam as determined by 
indicator cards taken from the steam cylinders. This method 
differs from that used in the previous articles, and gives results 
too small from the standpoint of hydraulic losses.' A discussion 
by Webber * of several tests shows that this efficiency increases 
with the lift as follows : 

Lift in feet, 5 15 30 100 170 270 

Efficiency, 0.30 0.45 0.65 0.83 0.91 0.88 

The highest value of 91 percent was obtained from a test of a 
Leavitt pumping engine having a duty of in 549 000 foot¬ 
pounds, and a capacity of 4400000 gallons per 24 hours; the 
duration of this test was 15.1 hours. 

Prob. 196 . In a test lasting 12 hours, 27502000 heat units were pro¬ 
duced under the boiler. The area of the plunger was 172 square inches, the 
length of the stroke was 18.9 inches, the number of single strokes was 76 000, 
and the leakage past the plunger packing was 5900 cubic feet. The pressure 
gage on the force pipe read 100 and the vacuum gage on the suction pipe 
read 9.3 pounds per square inch, the distance between the centers of these 
gages being 8 feet. The mean indicated horse-power of the steam cylinders 
was 128. Compute the discharge of the pump in cubic feet per second and 
its capacity in gallons per day. Compute the total pressure-head H. Com¬ 
pute the duty of the pumping engine. Compute the efficiency of the pump 
cylinders. 


Art. 197 . The Centrifugal Pump 

The centrifugal pump is the reverse of a turbine wheel, and 
any reaction turbine, when run backwards by power applied to 
its axle, will raise water through its penstock. The centrifugal 
pump, like the turbine, is of modern origin and development. 
A rude form, devised by Ledemour in 1730, consisted of an 
inclined tube attached by arms to a vertical shaft; the lower 


* Transactions American Society Mechanical Engineers, 1886, vol. 7, p. 602. 


The Centrifugal Pump. Art. 197 


521 


end of the tube being immersed, the water flowed from its upper 
end when the shaft was rotated. It was not, however, until 
about 1840 that the first true centrifugal pumps came into use, 
and they have since been perfected so as to be of great value in 
engineering operations, especially for low lifts. 


Fig. 197 shows the principle of the arrangement and action of 
the centrifugal pump. The power is applied through the 
axis A to rotate 
the wheel BB in 
the direction in¬ 
dicated by the ar¬ 
row. This wheel 
is formed of a 
number of curved 
vanes like those 
in a turbine wheel 
(Art. 174 ). The 
revolving vanes 
produce a partial 
vacuum, and this causes the water to rise in the suction pipe Dl) 
which enters through the center of the wheel case and delivers 
the water at the axis of the wheel. The water is then forced 
outward through the vanes and passes into the volute cham¬ 
ber CC, which is of varying cross-section in order to accom¬ 
modate the increasing quantity of water that is delivered 
into it, and all of which passes up the discharge pipe E. The 
rotation of the wheel hence produces a negative pressure at the 
upper end of the suction pipe and a positive pressure in the 
volute chamber, and the water rises in the pipes in the same 
manner as in those of a suction and force pump. The height of 
the suction lift cannot usually exceed about 28 feet. 



The parallelograms of velocity shown in Fig. 197 are the 
same as in the reaction turbine (Art. 174 ), and a similar notation 
will be used. The velocities of rotation of the inner and outer 
circumferences will be called u and ii \, the absolute velocities of 
the water as it enters and leaves the wheel are v 0 and v h and the 































522 


Chap. 16. Pumps and Pumping 


corresponding relative velocities are V and V\. The angles of 

entrance, approach, and exit are called </>, «, and / 3 , while 0 denotes 

the angle between v\ and U\. Let H 0 be the pressure-head at the 

top of the entrance pipe and Hi that at the foot of the discharge 

pipe, while 7 / 0 and hi are the heights of the suction and force lifts 

estimated downward and upward from the center of the wheel, 

and let h a be the height of the water barometer. Then from 

formula ( 162 ) 2 2 T/2 , 2 fTJ u v 

V 2 - u 2 - Vi + Ui 2 = 2 g {H 1 - Ho) 


and also from (31) 2 , not considering frictional resistances, 

Hi = h a -\- hi — — H<> = h a — ho~ — 

2 g 2 g 

Combining these equations, and replacing hi + /z 0 by h, where 
h is the total lift, the fundamental equation for the discussion of 
frictionless centrifugal pumps results. To introduce the fric¬ 
tional losses, however, h + h' should be used instead of h , where 
h' is the total head lost in all the hydraulic resistances. Then 

V 2 — Vi 2 - u 2 + ui 2 + vi 2 — Vo 2 = 2 g (h + h!) ( 197 ) i 

is the fundamental formula for the discussion of the centrifugal 
pump. Since there are no guides, the water enters the vanes 
radially, so that the approach angle a is a right angle, and hence 
1 2 = u 2 + Vo 2 . Also the parallelogram of velocities at exit gives 
V 2 = Ui 2 + vi 2 — 2UiViQ.os0. Inserting these values of V 2 and 
Pi 2 in ( 197 )i, it reduces to 

UiVi cos 0 = g(h + h') 

which is a necessary relation connecting u x and V\. 

A centrifugal pump must be run at a certain velocity in order 
to overcome the pressure-head h -f- h' by means of the velocity- 
head vi~/2g of the issuing water. Hence h + h f = Vi 2 /2g, and 
equating this to the value of h -f h! established by the above 
formula, there results m cos 0 = \v x . It hence follows from the 
parallelogram of velocities that Vi and u x must be equal. Then 
6 = 90° — hft, and _ 

«i = —^ or Ml= V 2g(//+70 
2 sin |/3 2 sin §/8 


( 197 ), 






The Centrifugal Pump. Art. 197 


523 


gives the required velocity of the outer circumference of the wheel. 
This velocity decreases as the exit angle /3 increases ] when (3 is 
very small, u x is very large ; when th e vanes are radial at the outer 
circumference, (3 is go and U\ = “x/g (h -\- h'). Hence the speed 
of the pump must increase with the square root of the pressure- 
head h + h!. Since Vi = q/a h where a x is the area of the exit 
orifices normal to Vi, the velocity is also Ui = q/2a x sin J/3, and 
therefore the discharge q increases directly with the speed. 

Since the speed must increase with the lift, and since the losses 
of head increase with the speed, it follows that the efficiency of 
the centrifugal pump in general decreases with the lift. This 
theoretic conclusion has been verified by practical tests. Webber, 
in his discussion cited in the last article, gives the following as 
the mean results derived from a number of experiments, the 
efficiency computed being the ratio of the work done by the pump 
to that obtained from indicator cards taken on the cylinders of 
the steam motor: 

Lift in feet, 5 10 20 40 60 

Efficiency, 0.56 0.64 0.68 0.58 0.40 

For a low lift the centrifugal pump has a hydraulic efficiency 
higher than these figures indicate, but, as in the case of the force 
pump, it is difficult to determine reliable values by numerical 
computations. 

The centrifugal pump possesses an advantage over the force pump 
in having no valves and in being able to handle muddy water, for even 
gravel may pass through the vanes without injuring them. The 
above figure represents the principle rather than the actual details 
of construction. Usually the suction pipe is divided into two parts 
which enter the axis upon opposite sides of the wheel, and the volute 
chamber is often made wider than the wheel case, thus forming what 
is called a whirlpool chamber, which prevents some of the losses of 
head due to impact. The vanes are sometimes curved in the oppo¬ 
site direction to that shown in the figure, as by so doing the angle 
is made larger and the speed of the pump is lessened, as is seen from 
formula ( 197 ) 2 . The theory of the centrifugal pump is, however, 
much less definite than that of the reaction turbine, and experiment 
is the best guide to determine the advantageous shape of the vanes. 



524 


Chap. 16. Pumps and Pumping 


Multiple stage centrifugal pumps for work against high heads 
are extensively used.*f 

Prob. 197 . A centrifugal pump lifts 120 cubic feet of water per minute 
through a discharge pipe having a diameter of 1 foot. The outer diameter 
of the wheel is 2 feet, the exit angle is go°. the number of revolutions per sec¬ 
ond is 60, and the water is lifted 18 feet. Compute the horse-power of the 
pump, and its hydraulic efficiency. 

Art. 198 . The Hydraulic Ram 

The hydraulic ram is an apparatus which employs the dynamic 
pressure produced by stopping a column of moving water to raise 
a part of this water to a higher level than that of its source. The 
principle of its action was recognized by Whitehurst in 1772,! 
but the credit of perfecting the machine is due to Montgolfier, 
who in 1796 built the first self-acting ram. It has since been 
widely used for pumping small quantities of water from streams 
to houses, but is not so well adapted to lifting a large quantity; 
many attempts have been made in this direction, some of which 
give promise of much usefulness. 

The principle of the action of the hydraulic ram is shown in 
Fig. 198 , where A is the reservoir that furnishes the supply, BCD 



the ram, AB the drive pipe which carries the water to the ram, 
DE the discharge pipe through which a part of the water is 
raised to the tank E. The ram itself consists merely of the waste 
valve B through which a part of the water from the drive pipe 

* Journal American Society of Mechanical Engineers, Jan. and March, 1910. 

f Journal Western Society of Engineers, April, 1910. 

\ Transactions Royal Society, 1775, vol. 65, p. 277. 











The Hydraulic Ram. Art. 198 


525 


escapes, and the air vessel D which has a valve C that allows 
water to enter it through BC, but prevents its return. The waste 
valve B is either weighted or arranged with a spring so that it 
will open when acted upon by the static pressure due to the head 
H. As soon as it opens the water flows through it, but as the 
velocity increases the dynamic pressure due to the motion of 
the column AB (Art. 157 ) becomes sufficiently great to close the 
valve B. Then this dynamic pressure opens the valve C and 
compresses the air in the air chamber or forces water up the dis¬ 
charge pipe. A moment later when equilibrium has obtained in 
the air vessel, the valve C closes and the air pressure maintains 
the flow for a short period in the discharge pipe, while the water 
in the drive pipe comes to rest. Then the waste valve B opens 
again, and the same operations are repeated. 

The algebraic discussion of the hydraulic ram is very difficult 
because it involves the time in which the waste valve closes and 
the law of its rate of closing. The investigation in Art. 157 , 
however, clearly shows that the operations above described will 
take place if the drive pipe is long enough to produce a dynamic 
pressure sufficient to close the waste valve. Let l be the length 
of that pipe, v the velocity in it, po the static unit pressure due 
to H, w the weight of a cubit unit of water, g the acceleration 
of gravity, and t the time in which the valve closes. Then, since 
there is no static pressure at the valve during the flow, the for- 
mula ( 157 ), gives p = 2wh/gt _ p<> 

which is a good approximation to the excess of dynamic pressure 
over the static pressure p 0 . It is seen that this excess p may be 
rendered very great by making l large and t small, and that its 
greatest value is p = wuv/g _ po 

in which u is the velocity of sound in water. It is rare, however, 
that a drive pipe is sufficiently long to furnish the excess dynamic 
pressure given by the last formula. 

The efficiency of the hydraulic ram is the ratio of the useful 
work done to the energy expended in the waste water. Let q 
be the quantity of water lifted per second through the height h 


526 


Chap. 16 . Pumps and Pumping 


from the level of the reservoir A to that of the tank E. Let Q 
be the discharge per second through the waste valve and H the 
height through which it falls, then the efficiency of the ram and 
its pipes is = = qjj^ 

6 wQH QH 


It is found by experiment that the efficiency decreases as the ratio 
h/H increases. Eytelwein found that e was 0.92 when h/H was 
unity, 0.67 when h/H was 5, and 0.23 when h/H was 20, but these 
values were probably derived by using a different formula for 
the efficiency. 


Experiments in 1890 at Lehigh University on a Gould ram No. 2, 
in which the waste valve made 55 strokes per minute, gave a mean 
efficiency of 35 percent. The length of the supply pipe was 38 feet 
and its fall 12 feet, the length of the discharge pipe 60 feet, and the lift 
h was 12 feet, so that the ratio h/H was unity. These experiments 
showed also that the efficiency increased as the number of strokes 
per minute was decreased by lessening the weight on the waste valve. 
The maximum quantity of water raised per minute, however, oc¬ 
curred with a heavier waste valve than that which gave the maximum 
efficiency. The efficiency was also found to increase as the length of 
the stroke of the w r aste valve decreased. 


The least possible fall in the drive pipe of the hydraulic ram is about 
ii feet and the least length of drive pipe about 15 feet. It is customary 
to make the area of the discharge pipe from one-third to one-fourth 
that of the drive pipe, and w r ith these proportions a fall of 10 feet will 
force water to a height of nearly 150 feet. A common rule of manu¬ 
facturers is that about one-seventh of the water flowing down the drive 
pipe may be raised to a height five times that of the fall in the drive 
pipe; this is a rough rule only, for the length of the discharge pipe 
is one of the controlling factors as well as its vertical rise. 

The Rife hydraulic engine is a water ram on a large scale, two or 
more being connected to the same discharge pipe, so that the flow in 
it is nearly continuous.* Three of these engines are said to raise 
864 000 gallons of water per day to an elevation of 150 feet, the fall in 
the drive pipe being 30 feet. The diameter of the drive pipe is 8 inches 
and that of the discharge pipe is 4 inches; the waste valve weighs 


* Engineering News, 1896, vol. 36, p. 429. 




Other Kinds of Pumps. Art. 199 527 

50 pounds, and it is provided with an adjusting lever in order that 
its effective weight may be regulated so as to cause the maximum 
discharge to be delivered. 

Prob. 198 . A hydraulic ram raises 32^ pounds of water in 5 minutes 
through a discharge pipe 60 feet long. The drive pipe is 38 feet long and the 
amount of water wasted in 5 minutes is 41^ pounds. The fall of the drive 
pipe is 12 feet and the vertical rise of the discharge pipe above the ram is 
24 feet. Compute the efficiency of the ram. 

Art. 199 . Other Kinds of Pumps 

The lift and force pumps described in Arts. 193 and 194 are 
called displacement pumps, because the volume of water lifted 
in one stroke is that displaced by the piston or plunger. If there 
be no leakage past the piston packing, and if no air is mingled 
with the water, the discharge in a given time may be very accu¬ 
rately determined by counting the number of strokes and multi¬ 
plying this number by the displacement in one stroke. On 
account of the reciprocating motion of the piston these forms 
are often called reciprocating pumps. There is always a loss of 
energy due to putting the piston into motion at the beginning of 
each stroke, and to avoid this many forms of rotary pumps have 
been devised; yet notwithstanding this loss the plunger force 
pump is probably the most efficient and economical of all kinds. 

A rotary or impeller pump is one in which the moving parts 
have a circular motion only, and the centrifugal pump described 
in Art. 197 is of this kind. Numerous other rotary pumps have 
been invented, but none is widely used except the centrifugal one. 
Fig. 199 # shows one where the moving parts consist of two wheels 
which are rotated in opposite directions as indicated by the 
arrows; this motion produces a partial vacuum whereby the 
water rises in the suction pipe D, and is then carried between the 
teeth and the case and forced up the discharge pipe E . Fig. 1996 
shows a form where the moving parts are two lobes in contact 
with each other and each in contact with the inclosing case. In 
the left-hand diagram the water rising in the pipe D is flowing 
toward the right, but a moment later the lobe B has assumed 


528 


Chap. 16. Pumps and Pumping 


the position shown in the right-hand diagram, and the water is 
imprisoned between the lobe and the case. An instant later the 
two lobes are forcing this water up the pipe E, while the water 
coming in at D is flowing to the left. The greatest objection to 




these pumps is that it is difficult to maintain close contact be¬ 
tween the case and the lobes or wheels, owing to wear, so that 
after being in use for some time there is much back leakage of 
water, and the capacity and efficiency of the pump are diminished. 
The only apparent advantage of the rotary pump is that it has 
no valves. Five rotary pumps of the type of Fig. 1995 were 
installed in 1902 at a pumping station near Chicago, the lobes 
or impellers being 4 feet long and the distance between their 
centers 2.7 feet; these pumps run at 100 revolutions per minute, 
and each has a capacity of 6000 cubic feet per minute under the 
total lift of about 8 feet.* 

The pumps thus far described, with the exception of the 
hydraulic ram, may be called mechanical pumps, because they 
act under energy communicated to them from motors. All 
mechanical pumps are reversible; that is, when the water moves 
in the opposite direction under a pressure-head, they become 
hydraulic motors. The reverse of the chain and bucket pump 
is the overshot or breast wheel, that of the suction and lift pump 
is the water-pressure engine, and that of the centrifugal pump is 
the turbine. The hydraulic ram does not operate under the ac¬ 
tion of a motor, and it does not appear to be reversible. 

* Engineering News, 1903, vol. 49, p. 172. 




















Other Kinds of Pumps. Art. 199 


529 


Pumps which have no moving parts and which operate through 
the action of air suction and dynamic pressure constitute another 
class which will now be briefly considered. Here belong the 
jet or ejector pumps which act largely through suction, and the 
injector pump used on locomotives. The latter produces a 
vacuum through the flow of steam, and cannot be discussed here, 
as it involves principles of thermodynamics. The fundamental 
principle, however, is indicated in Fig. 199 c, which shows the jet 
apparatus invented by James Thomson in 1850.* The water to 
be lifted is at C, and it rises by 
suction to the chamber B, from 
which it passes through the dis¬ 
charge pipe to the tank D. The 
forces of suction and pressure are 
produced by a jet of water issuing 
from a nozzle at the mouth of the 
discharge pipe, the nozzle being at 
the end of a pipe A B through 
which water is brought from a reservoir; or the water delivered 
from the nozzle may come from a hydrant or from a force pump. 
Let II be the effective head of the jet as it issues from the 
nozzle, ih the suction lift, and h 2 the lift above the tip of the 
nozzle; let q be the discharge through the nozzle and qi that 
through the suction pipe. Then, neglecting frictional resistances, 

qll = qhi + qi {hi ~b h 2 ) 
e = (qh 2 + qJh + qih 2 ) / qH 

It is found by experiments that the efficiency of this jet pump 
is very low, usually riot exceeding 20 percent, the highest effi¬ 
ciencies being for low ratios of hi + h 2 to H. This form of pump 
has, however, been found very convenient in keeping coffei dams 
and sewer trenches free from water, as it requires little or no atten¬ 
tion and has no moving parts to get out of order. 

Another class of pumps uses the pressure of air or of steam in 
order to elevate water. The idea of these pumps is old, yet it was not 
until 1875 that the steam pulsometer was perfected by Hall, while 

* Report of British Association, 1852, p. 130. 













530 


Chap. 16. Pumps and Pumping 


the air-lift pump of Frizell dates from 1880. The air-lift pump is now 
extensively used for raising water from deep wells, the compressed air 
being forced down a vertical pipe in the well tube and issuing from its 
lower end. As it issues, bubbles are formed in the entire column of 
water in the well tube, and being lighter than a column of common 
water, it rises to a greater height under the atmospheric pressure, 
assisted by the upward impulse of the bubbles to a slight extent. 
In this manner water having a natural level 50 feet or more below 
the surface of the ground may be caused to rise above that surface. 
It has been found in practice that for lifts of 15 to 50 feet from 

2 to 3 cubic feet of air are necessary for each cubic foot of water 
that is elevated. The efficiency of this form of pump is low, rarely 
reaching 30 percent, although a maximum of 50 percent has been 
claimed.* 

Among the many forms of pumps operating under the pressure 
of compressed air only the ejector pump used in the Shone system 
of sewerage can here be mentioned. The sewage from a number of 
houses flows to a closed basin, called an injector, in which it continues 
to accumulate until a valve is opened by a float. The opening of this 
valve allows compressed air to enter, and this drives out the sewage 
through a discharge pipe to the place where it is desired to deliver it. 
In the installation of this system of sewerage at the World’s Fair 
of 1893 in Chicago, there were 26 ejectors which lifted the sewage 
67 feet, the total pressure-head being about 108 feet. Vacuum methods 
of moving sewage have also been used in Europe, but these cannot 
compete in efficiency with those using compressed air. 

Prob. 199 . For Fig. 199 c let the diameter of the nozzle be 1 inch and 
that of the discharge pipe 4 inches. Let H be 64 feet, h x be 18 feet, //., be 

3 feet, and the discharge from the nozzle be 0.25 cubic feet per second. 
Compute the greatest quantity of water that can be lifted per second through 
the suction pipe, and the efficiency of the apparatus when doing this work. 


Art. 200 . Pumping through Pipes 

When water is pumped through a pipe from a lower to a higher 
level, the power of the pump must be sufficient not only to raise 
the required amount in a given time, but also to overcome the 
various resistances to flow. The head due to the resistances is 


* Journal of Association of Engineering Societies, 1900, vol. 25, p. 173. 


Pumping through Pipes. Art. 200 


531 


thus a direct source of loss, and it is desirable that the pipe 
should be so arranged as to render this as small as possible. 
The length of the pipe is usually much greater than the vertical 
lift, so that the losses of head in friction are materially higher 
than those indicated by the discussion of Art. 195 , where vertical 
discharge pipes were alone considered. 

Let w be the weight of a cubic foot of water and q the quantity 
raised per second through the height h , which, for example, may 

be the difference in level be- _ / _ ^ _ 

tween a canal C and a reser- y ! 

voir R , as in Fig. 200 a. The 
useful work done by the 
pump in each second is wqh. 

Let h' be the head lost in 
entering the pipe at the 
canal, h" that lost in friction in the pipe, and h' n all other losses 
of head, such as those caused by curves, valves, and by re¬ 
sistances in passing through the pump cylinders. Then the 
total work performed by the pump per second is 

k = wqh + wq {Ji + h" + h"') (200) i 

Inserting the values of the lost heads from Arts. 89 - 92 , this 
expression takes the form 

k = wqh + wq {^ni +/ ^ (200) 2 

in which v is the velocity in the pipe, l its length, and d its diameter. 
In order, therefore, that the losses of work may be as small as 
possible, the velocity of flow through the pipe should be low; 
and this is to be effected by making the diameter of the pipe 
large. The size of the pipe is here regarded as uniform from the 
canal to the reservoir; in practice the suction pipe is usually 
larger in diameter than the discharge pipe, in order that the suc¬ 
tion valves may receive an ample supply of water. 

For example, let it be required to determine the horse-power 
of a pump to raise i 200 000 gallons per day through a height of 









532 


Chap. 16. Pumps and Pumping 


230 feet when the diameter of the pipe is 6 inches and its length 
1400 feet. The discharge per second is 



1 200 000 
7.481 X 24 X 3600 


= 1.86 cubic feet, 


and the velocity in the pipe is 

1.86 


v 


= 9.47 feet per second. 


0.7854 Xo.5 ! 

The probable head lost in entering the pipe is, by Art. 89 , 


ti = 0.5 —■ = 0.5 X 1.39 = 0.7 feet. 

When the pipe is new and clean, the friction factor / is about 
0.020, as shown by Table 90 a; then the loss of head in friction 
in the pipe is, by Art. 90 , 


h" = 0.020 X X 1.39 = 77.8 feet. 

°-5 

The other losses of head depend upon the details of the pump 
cylinder and the valves; if these be such that 7^2 = 4, then 

ti" = 4 X 1.39 = 5.6 feet. 

The total losses of head hence are 


ti + ti T h " — 84.1 feet. 

The work to be performed per second by the pump now is 

k = 62.5 X 1.86 (230 + 84.1) = 36 510 foot-pounds, 

and the horse-power to be expended is 36 510/550 = 66.4. If 
there were no losses in friction and other resistances, the work 
to be done would be simply 

k = 62.5 X 1.86 X 230 = 26 740 foot-pounds, 

and the corresponding horse-power would be 26 740/550 = 48.6. 
Hence 17.8 horse-power is wasted in injurious resistances, or 
the efficiency of the plant is only 73 percent. 

For the same data let the 6-inch pipe be replaced by one 14 ' 
inches in diameter. Then, proceeding as before, the velocity of 
flow is found to be 1.74 feet per second, the head lost at entrance 





Pumping through Pipes. Art. 200 


533 


0.03 feet, the head lost in friction 1.13 feet, and that lost in other 
ways 0.19 feet. The total losses of head are thus only 1.35 feet, 
as against 84.1 feet for the smaller pipe, and the horse-power 
required is 48.9, which is but little greater than the theoretic 
power. The great advantage of the larger pipe is thus apparent, 
and by increasing its size to 18 inches the losses of head may be 
reduced so low as to be scarcely appreciable in comparison with 
the useful head of 230 feet. 

A pump is often used to force water directly through the mains 
of a water-supply system under a designated pressure. The work 
of the pump in this case consists of that required to maintain the pres¬ 
sure and that required to overcome the frictional resistances. Let 
h be the pressure-head to be maintained at the end of the main, 
and z the height of the main above the level of the river from which 
the water is pumped; then /q+z is the head II, which corresponds to 
the useful work of the pump, and, as before, 

k = wqH + wq (h! + h" -j- h' n ) 

To reduce the injurious heads to the smallest limits the mains should 
be large in order that the velocity of flow may be small. In Fig. 
2006 is shown a symbolic representation of the case of pumping into 
a main, P being the pump, C the 
source of supply, and DM the pres¬ 
sure-head which is maintained upon 
the end of the pipe during the 
flow. At the pump the pressure- 
head is AP , so that AD represents 
the hydraulic gradient for the pipe 
from P to M. The total work of 
the pump may then be regarded as 
expended in lifting the water from 
C to A, and this consists of three parts corresponding to the heads CM 
or z, MD or h x , and AB or h'- f- h n + h'", the first overcoming the force 
of gravity, the second maintaining the discharge under the required 
pressure, while the last is transformed into heat in overcoming fric¬ 
tion and other resistances. In this direct method of water supply 
a standpipe, AP, is often erected near the pump, in which the water 
rises to a height corresponding to the required pressure, and which 
furnishes a supply when a temporary stoppage of the pumping engine 



















534 


Chap. 16 . Pumps and Pumping 


occurs. This standpipe also relieves the pump to some extent from 
the shock of water hammer (Art. 157 ). 

Prob. 200 . Compute the horse-power of a pump for the following data, 
neglecting all resistances except those due to pipe friction: <7 = 1.5 cubic 
feet per second, which is distributed uniformly over a length ^=3000 feet 
(Art. 104 ), the remaining length of the pipe being 4290 feet; d = 10 inches, 
/?i = 75.8 feet, and 2 = 10.6 feet. 

Art. 201 . Pumping through Hose 

In Art. 109 the flow of water through fire hose was briefly 
treated and the friction factors given for different kinds of hose 
linings. It was shown that the loss of head in a long hose line 
becomes so great, even under moderate velocities, as to consume 
a large proportion of the pressure exerted by the hydrant or 
steamer. As another example, let the pressure in the pump of 
the fire engine be 122 pounds per square inch, corresponding 
to a head of 281 feet, and let it be required to find the pressure- 
head in 2j-inch rough rubber-lined cotton hose at 1000 feet dis¬ 
tance, when a nozzle is used which discharges 153 gallons per 
minute, the hose being laid horizontal. The discharge is 0.341 
cubic feet per second, which gives a velocity of 10.o feet per sec¬ 
ond in the hose. Hence by ( 90 ) the loss of head in friction is 
231 feet, so that the pressure-head at the nozzle entrance is only 
50 feet, which corresponds to about 22 pounds per square inch. 
The remedy for this great reduction of pressure is to employ a 
smaller nozzle, thus decreasing the discharge and the velocity 
in the hose ; but if both head and discharge are desired, they may 
. be obtained either by an increase of pressure at the steamer or 
by the use of a larger hose. 

Another method of securing both high velocity-head and 
quantity of water is by the use of siamesed hose lines, and this 
is generally used when large fires occur. This method consists 
in having several lines of hose, generally four, lead from the 
steamer to a so-called Siamese connection, from which a short 
single line of hose leads to the nozzle. In Fig. 201 the pump 
or fire steamer is represented by A, the Siamese joint by B , the 
nozzle entrance by C, and the nozzle tip by D. From A let n 


Pumping through Hose. Art. 201 


535 


lines of hose, each having the length h and the diameter d h lead 
to B ; and from B let there be a single line of length l 2 and diam¬ 
eter d 2 leading to the nozzle which has the diameter D. The 
hydraulic gradient (Art. 99 ) is shown by abcD, the pressure-heads 


a 


'li 




.-v C 

I \ 

• \ 

•.>1 


B 


C D 


Fig. 201. 


at A, B, C being represented by Aa, Bb, Cc. Let h be the pres¬ 
sure-head on the nozzle tip or the difference of the elevations of 
the points a and D. It is required to deduce a formula for the 
velocity at the nozzle tip and to determine the pressure-heads 
at B and C. 


This case is one of diversions, already treated in Art. 105 , 
and the same principles may be applied to its solution. Neg¬ 
lecting losses in entrance, in curvature, and in the Siamese joint, 
the total head h is expended in friction in the hose lines and in 
the nozzle, or _ _ L 7 „2 . . L 7Jo 2 _ T y 2 


/ _ r ^1 | r 1'2 V 2 

h -J1 - \-J 2 — 

di 2 g do 2g 


+ 


d 2 2 g 


in which V\ and v 2 are the velocities in the lines h and l 2 , and V 
is that from the nozzle, while d is the coefficient of velocity of 
the nozzle (Art. 83 ). The first term of the second member is 
the head lost between A and B , and the algebraic expression for 
this is independent of the number of hose lines between those 
points; the velocity Vi in these hose lines depends, indeed, upon 
their number, but the hydraulic gradient ab is the same for each 
and all of them. The law of continuity of flow (Art. 31 ) gives, 
however, nd 2 V{ = ^ = jyi y 


and, taking from these the values of V\ and i’-> in terms of V and 
inserting them in the expression for there results 


Mi(D 

n’-di dj d« \d->) C\ 


(201) 















536 


Chap. 16. Pumps and Pumping 


from which the velocity V and the velocity-head V 2 /2g can be 
computed, while the discharge is given by q = \^D 2 V. The 
pressure-head h 2 at the nozzle entrance and the pressure-head hi 
at the Siamese joint may then be found from 

MY + nz? 

.di\d 2' Ci 2 j2g 

and, as a check, the latter should equal h minus the drop of the 
hydraulic gradient between a and h. 

This discussion shows that, by increasing the number n , the 
loss of head between A and B may be made very small, the effect 
being practically the same as that of moving the steamer to B 
and using but a single hose line / 2 - As a numerical example, 
let h = 230.4 feet, h = 500 feet, h = 60 feet, di = d 2 = 2.5 inches, 
D — 1 inch, and Ci = 0.975. Then, taking/as 0.03, the computed 
results for different values of n are as follows, V being in feet per 
second, V 2 /2g in feet, and q in gallons per minute. It is seen that 


n — 1 

2 

3 

4 

6 

00 

V = 68.9 

92.2 

99.8 

10.3 

105 

107 

V 2 /2 g = 73.7 

132 

155 

165 

173 

180 

q = 169 

226 

244 

252 

258 

263 


for four lines the velocity-head is more than double that for a 
single line and that the discharge is 50 percent greater. With 
more than four lines the velocity-head and discharge increase 
slowly, and for n — 00 they are practically the same as for n = 10. 
The number of hose lines generally used is four, since the slight 
advantage of more lines is not sufficient to warrant their use. 

Many other interesting problems relating to hose lines may 
be solved by using the same principles. If there are four lines of 
hose between the pump and the Siamese joint, three having the 
diameter d\ and one having the diameter d, it can be shown that 
the formula (201) applies, provided n be replaced by 3 + (d/d\)i. 
For instance, if d be 3 inches and d\ be 2J inches, this makes n 2 
about 19. In deducing this expression for n it is assumed that 
the friction factors are the same for both sizes of hose, although 
in strictness the smaller hose has the higher value of /. 






Pumping through Hose. Art. 201 


537 


Another case is where two of the hose lines between A and 
B have the diameter d x and the length l x , while the two other lines 
are of the length l + h, the length l having the diameter d and 
the length / 3 the diameter d 3 . Here the principles regarding com¬ 
pound pipes (Art. 100 ) are also to be regarded, and formula ( 201 ) 
applies likewise to this case, if n be computed from 

in which e represents f(l/d), while e x and e 3 represent f\(l\/d\) and 
fz{h/d 3 ) respectively. For instance, if h = ioo, / 3 = ioo, and 
/ = 50 feet, while d x = d 3 = 2\ inches and d = 3 inches, then 
the value of n 2 is about 21, so that this arrangement is more effec¬ 
tive than that of the preceding paragraph. 

In the deduction of the above formulas losses of head at entrance 
and in the Siamese joint have not been regarded, and it is unnecessary 
to consider these when the hose lines are long. For lines less than 100 
feet in length the losses of head at entrance may be taken into account 
by adding the term o.$(D/di) 2 /n 2 to the denominator of ( 201 ). The 
loss of head due to the Siamese joint may, in the absence of experi¬ 
mental data, be approximately accounted for by adding about 0.02 
to that denominator, thus considering its influence about one-half 
that of the nozzle. In a case like that of the last paragraph, where the 
length l in two of the hose lines is nearest the pumps, the values of 
e and e x may be increased by 0.5 in order to introduce the influence 
of the entrance heads. Errors of 5 percent or more are liable to occur 
in computations on pumping through short hose lines. 

Prob. 201 #. Three hose lines run from a pump to a Siamese connec¬ 
tion, each being 500 feet long and 2 \ inches in diameter, and from the Siamese 
one line 50 feet long and 2\ inches in diameter leads to a i|-inch nozzle hav¬ 
ing a velocity coefficient of 0.96. When the pressure at the pump is 100 
pounds per square inch, what is the discharge from the nozzle and the veloc¬ 
ity-head of the jet ? What friction heads are lost in the hose and nozzle ? 

Prob. 201 b. In a fire-engine test made in 1903, the lengths l x and l 2 
were 50 feet, the length l was 12 feet, and U was zero, as the nozzle was at¬ 
tached directly to the Siamese joint. The diameter d x was 3 inches, while 
d and d z were 2§ inches, and D was 2 inches. The pressure gage on the 
steamer read 90, while one on the Siamese joint read 63 pounds per square 
inch. Compute the pressure-head at the Siamese joint. 




538 


Chap. 16. Pumps and Pumping 


Prob. 201 c. What is the efficiency of a bucket pump which lifts 2000 
liters of water per minute through a height of 3.5 meters with an expenditure 
of 2.5 metric horse-powers? 

Prob. 201 d. When the height of the mercury barometer is 760 milli¬ 
meters, water at a temperature of o° centigrade is raised by suction in a per¬ 
fect vacuum to a height of 10.33 meters (Art. 193 ). Under the same at¬ 
mospheric pressure, how high can it be raised when the temperature is 32 0 
centigrade ? 

Prob. 201 c. What metric horse-power is required to raise 4 000 000 
liters per day through a height of 75 meters when the diameter of the pipe 
is 20 centimeters and its length 500 meters ? 

Prob. 201 /. The calorie is the metric thermal unit, this being the energy 
required to raise one kilogram of water one degree centigrade when the tem¬ 
perature of the water is near that of maximum density. How many calories 
are equivalent to 1 000000 British thermal units? 


Hydraulic-Electric Analogies. Art. 202 


539 


APPENDIX 

Art. 202. Hydraulic-Electric Analogies 

It is well known that there are certain analogies between the 
flow of water in pipes and that of the electric current in wires, 
and some of these will here be briefly explained from a hydraulic 
point of view. The electric analog of a water pump is the dynamo, 
both being driven by mechanical power and both transforming it 
into other forms of energy. The analog of a water wheel is the 
electric motor, each of which delivers mechanical power by virtue 
of the energy transmitted to it through the water pipe or electric 
wire. While the water is flowing from the pump to the wheel 
much of its energy is lost in overcoming frictional resistances, 
whereby heat is produced; while the electricity is flowing from 
the dynamo to the electric motor some of its energy is lost in 
overcoming molecular resistances, whereby heat is produced. 
The steady flow of water corresponds to the continuous flow of 
electricity in one direction, or to the direct current, and the fol¬ 
lowing discussion compares hydraulic phenomena with those of 
the direct electric current. The phenomena of the alternating 
current have also certain hydraulic analogies in the flow of 
water, but these will not be discussed here. 

Let q represent electric current, R the electric resistance of a 
wire of length /, cross-section a , and diameter d , and p the electro¬ 
motive force under which the current is pushed through the wire. 
Then Ohm’s law gives, if s is the specific resistance of the material 

of the wire, ; / 

p = Rq = s-q = A - q ( 202 )! 

a d z 

in which A is a constant depending only on the material of the 
wire. This equation shows that the electric pressure p varies 


540 


Appendix 


directly with the length of the wire, inversely as the square of 
its diameter, and directly as the current. By increasing the length 
of the wire or by decreasing its diameter, the electromotive force 
required to maintain a given electric current is increased. Sim¬ 
ilarly in a water pipe the friction-head required to maintain a 
given discharge increases directly as the length of the pipe, and 
is greater for a small pipe than for a large one (Art. 90 ). 

In Art. 105 it was pointed out that the distribution of water 
flow among several diversions of a pipe follows laws analogous to 
those of the electric current. It was there shown that the dis¬ 
charge q divides between the diversions inversely as their resist¬ 
ances, provided V/ 7/^ 5 be taken as the measure of resistance. In 
electric flow the direct current is the analog of the discharge in 
the water pipe, but Ohm’s law shows that the resistance is the 
simpler quantity f'l/d 2 . The hydraulic analog of electro-motive 
force is often taken to be the lost friction-head or its corresponding 
unit pressure, and this will be followed here. The loss in water 
pressure is represented by the hydraulic gradient (Art. 99 ), and 
the loss in electric pressure is often represented in a similar way, 
the gradient being a straight line in both cases. 


In order to make an algebraic comparison of the two phenomena, 
take the expression for friction-head in ( 90 ) and replace h" by p/w, 
where p is the loss of unit pressure in the length l, and w is the weight 
of a cubic unit of water; also replace v by q/a, and a by rd 2 . Then 
formula ( 90 ) becomes 



^ 1 2 = B L q 2 

g d* H d bH 


(202) 2 


in which the constant B depends upon the roughness of the surface 
and the force of gravity. Accordingly the lost pressure varies di¬ 
rectly as the length of the pipe, inversely as the fifth power of its 
diameter, and directly as the square of the discharge. 

Thus, in the case of a single water pipe or electric wire, 


for electric flow 
for hydraulic flow 


P = A J 2 ( 1 

a* 

p = B — q i 
F dP H 




Hydraulic-Electric Analogies. Art. 202 


541 


If each of these flows be divided among n diversions, as in Fig. 201 , 
the expressions for the pressure become 

for electric flow p = q 

nd 2 


for hydraulic flow 



Bl 9 

- Q 

n 2 d^ 


so that the drop of the gradient is far more rapid in the latter case; 
thus, when n is 3, the electromotive force for three wires is one-third 
of that for a single wire, but the hydraulic pressure for three pipes is 
one-ninth of that for a single pipe. 

The conclusion to be derived from this comparison is that the anal¬ 
ogies between hydraulic and electric flow are rough ones and cannot 
embrace all the quantities involved. The only perfect analogy is that 
p varies directly as l ; the analogy between hydraulic discharge and 
electric current is perfect only as regards its distribution between 
branches or diversions; the analogy between hydraulic and electric 
resistance is an imperfect one that is liable to lead to confusion. Al¬ 
though a decrease in size of the pipe or wire causes an increase in re¬ 
sistance, the law of increase is quite different in the two cases. If 
hydraulic resistance be defined as in Art. 105 , then the lost pressure 
p is not proportional to resistance, but to its square root, while the 
lost electric pressure p varies directly as electric resistance. 

For the viscous flow of water in pipes (Art. 110 ), where the resist¬ 
ances are those of sliding friction only, 



AWC\ l TJ l 

V^ =i V’ 


which shows that the lost pressure is proportional to q as in Ohm’s 
law, so that the analogy is closer than in the common motion of water, 
where the greater part of the loss is due to impact. The resistance, 
however, varies inversely as the square of the area of the pipe, while 
in electric flow it varies inversely as the first power of the area. Thus 
this analogy breaks down, as all analogies connecting electric and me¬ 
chanical phenomena are found to do sooner or later.* 

There are also analogies between the economic problems of elec¬ 
tricity and those of hydraulics. For a wire line for the electric trans¬ 
mission of power, let C be the annual expenditure in interest and sink- 


* Heavyside, Electromagnetic Theory (London, 1894), vol. 1, p. 232. 





542 


Appendix 


ing fund charges on account of the cost of the wire and D be the annual 
loss on account of the energy wasted in heating the wire, both for a 
wire of diameter unity. Then the total annual loss is Cd 2 + D/d 2 , 
and this is a minimum when D/d 2 equals Cd 2 ; that is, the size 
of the wire which gives the greatest economy is such that the annual 
value of the energy lost in heat equals the annual expenditure on the 
cost of the wire line. In a similar manner, let C and D represent the 
same quantities for a pipe line carrying water to a power plant, both 
for a pipe of diameter unity. Then, since the thicknesses of pipes 
vary as their diameters and their costs as the squares of the diame¬ 
ters, Cd 2 + D/d b is the total annual loss, and this is a minimum when 
D/d b equals § Cd 2 ; that is, the size of pipe which gives greatest econ¬ 
omy is such that the annual value of the energy lost in friction equals 
two-fifths of the annual expenditure on the cost of the pipe line.* 

Prob. 202 . A copper wire having a specific resistance of 0.0000016 
ohms is one centimeter in diameter. A steel rail having a specific resistance 
of 0.0000145 ohms has a section area of 54.8 square centimeters. A certain 
transmission line consists of 9 kilometers of the copper wire and 3 kilometers 
of the steel rail. Compute the loss in voltage required to maintain a direct 
current of 150 amperes. If the pressure at the beginning of the line is 2500 
volts and the rail is at the middle of the line, draw the electric gradient. 

Art. 203 . Miscellaneous Problems 

The following problems introduce subjects that have not 
been specifically treated in the preceding pages. Teachers who 
wish to offer prize problems to their classes may perhaps find 
some of these suitable for that purpose. 

Prob. 203 a. A wooden water tank 18 feet in diameter and 24 feet high 
is to be hooped with iron bands which may be safely spaced 6 inches apart 
at the middle of the height. How far apart should they be spaced at the 
bottom ? 

Prob. 203 b. A house is 60 feet lower than a spring A and 30 feet higher 
than a spring B. A pipe from A to the house runs near B. Explain a 
method by which the water from B can be drawn into the pipe and be deliv¬ 
ered at the house. 

Prob. 203 c. A river having a width of 300 feet on the surface, a cross- 
section of 1800 square feet, a hydraulic radius of 5.3 feet, and a slope of 1 
on 10 000, discharges 10 400 cubic feet per second. If it be frozen over to 
the depth of one foot, what will be its discharge ? 

* Adams, Proceedings American Society of Civil Engineers, May, 1907. 


Miscellaneous Problems. Art. 203 


543 


Prob. 203d. From a pumping station water is forced by direct pressure 
through a compound pipe, consisting of 7500 feet of 14-inch pipe, 4100 feet 
of 12-inch pipe, and 780 feet of 8-inch pipe, to a 6-inch pipe on which there 
are three hydrants A, B, and C. A is 133 feet from the end of the 8-inch 
pipe and 115 feet above the gage at the pumping station; B is 433 feet from 
the end of the 8-inch pipe and 135 feet above the gage ; C is 733 feet from the 
end of the 8-inch pipe and 125 feet above the gage. To each of these hy¬ 
drants is attached 50 feet of 2^-inch rubber-lined hose with a i-inch smooth 
nozzle at the end. When the gage at the pumping station reads 175 pounds 
per square inch, to what heights will the three streams be thrown from the 
three nozzles ? 

Prob. 203c. When a body falls vertically in water, its velocity soon be¬ 
comes constant. For a smooth sphere an approximate formula for this veloc¬ 
ity is 2gd(s — 1), in which d is the diameter of the sphere and x its spe¬ 
cific gravity. Compute the velocity v for a sphere having a diameter of 0.001 
feet and a specific gravity of 1.25. 

Prob. 203/. The velocity with which water flows through a sand filter 
bed varies directly as the head (Art. 110). If V is the velocity in meters 
per day, d the effective size of the sand grains in millimeters, h the head, 
/ the thickness of the sand bed, and t the centigrade temperature, 

V = 1000 (0.70 + 0.03/) ( h/l)d 2 

is the formula deduced by Hazen.* When t = $2°.4 centigrade, ^ = 0.4 
millimeters, 1 = 4 feet, and h = 0.4 feet, find how many million gallons per 
day will pass through one acre of filter beds. 

Prob. 203g. A bent U tube of uniform size is partly filled with water. 
Let the water in one leg be depressed a certain distance, causing that in the 
other to rise the same distance. When the depressing force is removed, the 
water oscillates up and down in the legs of the tube, the times of oscillation 
being isochronous. If l be the entire leng th of the water in the tube, show 
that the time of one oscillation is 7r V//2g. If the legs are inclined to the 
hori zontal at the angl es 6 and <f>, show that the time of one oscillation is 
7r V//g (sin# + sin<£). 

Prob. 203//. The bottom of a canal has the width 2 b, and it is desired 
to shape the banks so that the hydraulic radius of the cross-section may be 
constant. Show that the equation of the curve is 

y = r loge (x + V# 2 — r 2 ) ' (b + Vfr 2 — r 2 ) 
in which y is the depth of the water, x the half width of the water surface, and 
r the constant hydraulic radius. 

Prob. 203/. A river having a slope of 1 on 2500 runs due east. A line 
drawn due north at a point A on the river strikes at B, 5000 feet from A, 

* Report Massachusetts State Board of Health, 1892, p. 553. 







544 


Appendix 


the edge of a large swamp which it is desired to drain. The level of the water 
in this swamp is 0.5 feet below the river surface at A, and it is desired to 
lower that level 1.5 feet more. For this purpose a ditch is to be dug run¬ 
ning from A in a straight line on a uniform slope until it joins the river at 
a point C eastward from A. The discharge of this ditch, in order to properly 
drain the swamp, will be 25 cubic feet per second, its side slopes are to be 1 
on 1, the mean velocity is not to exceed 2.5 feet per second, and the coeffi¬ 
cient c in the Chezy formula is estimated at 70. Find the length and width 
of the most economical ditch. 

Art. 204 . Answers to Problems 

Below will be found answers to some of the problems given 
in the preceding pages, the numbers of the problems being placed 
in parentheses. In general it is not a good plan for a student to 
solve a problem in order to obtain a given answer. One object 
of solving problems is, of course, to obtain correct results, but 
the correctness of those results should be established by methods 
of verification rather than by the authority of a given answer. 
It is more profitable that a number of students should obtain 
different answers to a problem and engage in a discussion as to 
the correctness of their solutions than that all discussion should 
be stopped because a certain answer is given in the text. How¬ 
ever satisfactory it may be to know in advance the result of the 
solution of an exercise, let the student bear in mind that after com¬ 
mencement day answers to problems will not be given. 

(1) One horse-power. ( 3 ) 147.2 pounds. ( 4 ) See Table 4 . ( 7 ) See 

Index. (8) 29.56 inches. ( 96 ) 9.54 kilograms per square centimeter. 
( 9 d) 5575 kilograms. ( 12 ) 40.6, 1.56, 2.65. ( 15 ) 28300 pounds. ( 17 ) 

4.01 feet. ( 206 ) 3.07. (20c) 2945 kilograms. (21) 56.9 feet per second. 

( 25 ) ^ = 32.1 feet per second. ( 27 ) 19.3 pounds. ( 32 ) 24.9 seconds. 
( 33 c) 0.73. ( 35 ) 1.96 and 166 cubic feet. ( 36 ) 0.017 inches. ( 37 ) 1.15 

feet. ( 39 ) v = 4.00 feet per second. ( 41 ) See Engineering News, May 4, 
1911. ( 45 ) c = 1.06. ( 48 ) c = 0.605. ( 49 ) 17.2 feet. ( 50 ) 10.5 cubic feet 
per second. ( 51 ) 0.034 cubic feet per second. ( 55 ) 103. ( 59 a) c L = 0.98. 

( 60 ) 0.361 feet per second. ( 62 ) 0.0109 feet. ( 67 ) 7.10 and 6.97 cubic 
feet per second. ( 71 ) 0.74 percent. ( 72 ) 0.581. ( 72 a) 1.30 centimeters. 

( 75 ) 0.126 feet. ( 76 ) 0.13 and 7.60 feet. ( 77 ) 0.28 feet. ( 78 ) c = 0.90 
and/q = 0.70. ( 80 ) c = 0.802. (SI) 6.67 feet. ( 83 ) 0.963. ( 84 ) 1.06. 

( 89 ) 0.29 feet. ( 95 ) 3.06 and 4.94 inches. ( 98 ) About 6 cubic feet per 
second. ( 112 ) 2.8 feet. ( 114 ) 4.4 feet. ( 115 ) 7.32 feet per second. ( 116 ) 


Mathematical Tables. Art. 205 


545 


1.28 X 0.64 feet. ( 118 ) 57 400000 gallons. ( 120 ) d = 3.09 feet. ( 1276 ) 
0.48 meters. ( 129 ) 546 cubic feet per second. ( 132 ) 1.76 feet per second. 
( 134 ) 760 cubic feet per second. ( 140 ) d x = 12.5 feet. ( 14 k) H = 0.41 
meters. ( 145 ) 0.9. ( 146 ) 13.5 horse-powers. ( 147 ) 1.32 horse-powers. 

( 148 ) 257 feet. ( 149 ) 35.4 percent. ( 15 k) 18400 kilowatts. ( 152 ) 
3.96 gallons. ( 155 ) About 120 pounds. ( 159 ) 34.5 feet per second. 
( 162 a) e = 0.83. ( 164 ) From 48 to 50 horse-powers. ( 165 ) 13.6. 

( 171 a) 30.1 kilowatts. ( 172 ) 16 feet. ( 175 ) 4.117 and 4.120. ( 178 ) 

167. ( 182 e) 27.0 cubic meters. ( 183 ) 743 horse-powers. ( 185 ) 1530 

horse-powers. ( 19 k) r = n. 6 meters. ( 198 ) e = 0.78. ( 200 ) 17.8 horse- 

. powers. (20k) g\ meters. 

Evolvi varia problemata. In scientiis enim ediscendis prosunt exempla 
magisquam praecepta. Qua de causa in his fusius expatiatus sum. — Newton. 

Art. 205 . Mathematical Tables 

Tables A, B, C, D give constants often needed in computations. 

Table E gives squares of numbers from 1.00 to 9.99, the arrange¬ 
ment being the same as»that of the logarithmic table. By properly 
moving the decimal point, four-place squares of other numbers are 
also readily taken out. For example, the square of 0.874 is 0.7639, 
and that of 87.4 is 7639, correct to four significant figures. 

Table F gives areas of circles for diameters ranging from 1.00 to 
9.99, arranged in the same manner, and by properly moving the deci¬ 
mal point, four-place areas for all circles can be found. For in¬ 
stance, if the diameter is 4.175 inches, the area is 13.69 square inches; 
if the diameter is 0.535 feet, the area is 0.2248 square feet. 

Table G gives trigonometric functions of angles and Table H 
the logarithms of these functions. The term “arc” means the length 
of a circular arc of radius unity, while “coarc” is the complement of 
the arc, or a quadrant minus the arc. If 6 is the number of degrees 
in any angle, the value of arc# is 7 t#/i 8 o. 

Table J gives four-place common logarithms of numbers, and 
these are of great value in hydraulic computations (Art. 8). Table 
K, taken from the author’s “Elements of Precise Surveying and 
Geodesy,” gives nine-place constants and their logarithms. 

For other tables used in hydraulic computations see American 
Civil Engineers’ Pocket Book (New York, 1912). Barlow’s Tables 
(London, 1907) give eight-place values of squares, cubes, square 
roots, cube roots, and reciprocals of numbers from 1 to 10 000. 


546 


Appendix 


Table A. Fundamental Hydraulic Constants 


English Measures 


Name 

Symbol 

Number 

Logarithm 

Pounds of water in one cubic foot 

w 

62.5 

1-7959 

Pounds of water in one U. S. gallon 
Pounds per square inch due to one 

w/ 7.481 

8-355 

0.9220 

atmosphere 

Pounds per square inch due to one foot 


14.7 

1.1673 

of head 

Feet of head for pressure of one pound 

w/144 

0-434 

1.6375 

per square inch 

144 /w 

2.304 

0.3625 

Cubic feet in one U. S. gallon 

231/1728 

0.1337 

1.1261 

U. S. gallons in one cubic foot 
Acceleration of gravity in feet per 

1728/231 

W 

00 

0.8739 

second per second 

s_ 

32.16 

1.5073 


V' 2 g 

8.020 

0.9042 


1^2 g 

5-347 

0.7281 


l/2£ 

0.01555 

2.1916 


\ 7 T V 2 f* 

6.299 

0-7993 


Table B. Fundamental Hydraulic Constants 

Metric Measures 


Name 

Symbol 

Number 

Logarithm 

Kilograms of water in one cubic meter 

w 

IOOO 

3.0000 

Kilograms of water in one liter 

w / 1000 

I 

0.0000 

Kilograms per square centimeter due to 




one atmosphere 


1-033 

O.OI42 

Kilograms per square centimeter due to 




one meter head 

w/ 10000 

0.1 

I. OOOO 

Meters of head for pressure of one kilo- 




gram per square centimeter 

10000/ zv 

10 

I. OOOO 

Cubic meters in one liter 

I / 1000 

0.001 

3.0000 

Liters in one cubic meter 

1000/1 

1000 

3.0000 

Acceleration of gravity in meters per 




second per second 

g 

9.800 

O.991 2 


V2g 

4.427 

O.6461 


f 

2.951 

O.4700 


I / 2 g 

0.05104 

2.7077 


\ TT ^ 2 g 

3-477 

O.5412 




















Mathematical Tables. Art. 205 


547 


Table C. Metric Equivalents of English Units 


English Unit 

Metric Equivalent 

Logarithm 

i Inch 

2.5400 centimeters 

0.40483 

i Foot 

0.3048 meters 

I.48402 

i Square Inch 

6.4520 square centimeters 

0.80969 

i Square Foot 

0.09290 square meters 

2.96803 

i Cubic Foot 

0.02832 cubic meters 

2.45209 

i U. S. Gallon 

3.7854 liters 

O.57812 

i Imperial Gallon 

4.5438 liters 

O.65742 

i Pound 

0.4536 kilograms 

I.65667 

i Pound per Square Inch 

0.07030 kilograms per square centi- 



meter 

2.84697 

i Pound per Cubic Foot 

16.017 kilograms per cubic meter 

I.20457 

i Foot-pound 

0.1383 kilogram-meters 

I.I4069 

i Horse-power 

1.0139 cheval-vapeur 

O.O0599 

Fahrenheit 

Centigrade temperature 


Temperature F° 

C°=f(F°—32 0 ) 



Table D. English Equivalents of Metric Units 


Metric Unit 

English Equivalent 

Logarithm 

i Centimeter 

0.3937 inches 

I- 595 I 7 

1 Meter 

3.2808 feet 

0.51598 

1 Square Centimeter 

0.1550 square inches 

1.19031 

1 Square Meter 

10.764 square feet 

I- 03 I 97 

1 Cubic Meter 

35.314 cubic feet 

I- 5479 I 

1 Liter 

0.2642 U. S. gallons 

T.42188 

1 Liter 

0.2201 imperial gallons 

1.34258 

1 Kilogram 

2.2046 pounds 

0-34333 

1 Kilogram per Square 



Centimeter 

14.224 pounds per square inch 

1-15303 

1 Kilogram per Cubic 



Meter 

0.06244 pounds per cubic foot 

2-79543 

1 Kilogram-meter 

7.2329 foot-pounds 

0.85931 

1 Cheval-vapeur 

0.9863 horse-powers 

T.99041 

Centigrade 

Fahrenheit temperature 


Temperature C° 

F°= 32 °+i .8 C° 





















548 


Appendix 


Table E. Squares of Numbers 


n 

01234 

56789 

Diff. 

1.0 

1.000 1.020 1.040 1.061 1.082 

1.103 1.124 1.145 1.166 1.188 

22 

1.1 

1.210 1.232 1.254 1.277 1.300 

1.323 1.346 1.369 i- 39 2 1.4*6 

24 

1.2 

1.440 1.464 1.488 1.513 1.538 

1.563 1.588 1.613 1-638 1.664 

26 

1.3 

1.690 1.716 1.742 1.769 1.796 

1.823 1.850 1.877 1.904 1.932 

28 

1.4 

1.960 1.988 2.016 2.045 2.074 

2.103 2.132 2.161 2.190 2.220 

30 

1.5 

2.250 2.280 2.310 2.341 2.372 

2.403 2.434 2.465 2.496 2.528 

32 

1.6 

2.560 2.592 2.624 2.657 2.690 

2.723 2.756 2.789 2.822 2.856 

34 

1 -7 

2.890 2.924 2.958 2.993 3.028 

3.063 3.098 3.133 3.168 3.204 

36 

1.8 

3.240 3.276 3.312 3.349 3.386 

3.423 3.460 3.497 3.534 3-572 

38 

1.9 

3.610 3.648 3.686 3.725 3.764 

3.803 3.842 3.881 3.920 3.960 

40 

s 

2.0 

4.000 4.040 4.080 4.121 4.162 

4.203 4.244 4.285 4.326 4.368 

42 

2.1 

4.410 4.452 4.494 4.537 4-580 

4.623 4.666 4-709 4 - 75 2 4-796 

44 

2.2 

4.840 4.884 4.928 4.973 5.018 

5.063 5.108 5.153 5.198 5- 2 44 

46 

2-3 

5.290 5.336 5.382 5.429 5.476 

5-523 5-570 5.617 5.664 5.712 

48 

2.4 

5.760.5.808 5.856 5.905 5.954 

6.003 6.052 6.101 6.150 6.200 

50 

2.5 

6.250 6.300 6.350 6.401 6.452 

6.503 6.554 6.605 6.656 6 708 

52 

2.6 

6.760 6.812 6.864 6.917 6.970 

7.023 7.076 7.129 7.182 7.236 

54 

2.7 

7.290 7.344 7.398 7.453 7.508 

7.563 7.618 7.673 7.728 7.784 

56 

2.8 

7.840 7.896 7.952 8.009 8.066 

8.123 8.180 8.237 8.294 8.352 

58 

2.9 

8.410 8.468 8.526 8.585 8.644 

8.703 8.762 8.821 8.880 8.940 

60 

3 -° 

9.000 9.060 9.120 9.181 9.242 

9.303 9.364 9.425 9.486 9.548 

62 

3 -i 

9.610 9.672 9.734 9-797 9-86o 

9.923 9.986 10.05 10.11 10.18 

6 

3-2 

10.24 10.30 10.37 10.43 IO -5° 

10.56 10.63 10.69 10.76 10.82 

7 

3-3 

10.89 10.96 11.02 11.09 n.i6 

11.22 11.29 11.36 11.42 11.49 

7 

3-4 

11.56 11.63 ii-7o 11.76 11.83 

11.90 11.97 12.04 12.11 12.18 

7 

3-5 

12.25 12.32 12.39 12.46 12.53 

12.60 12.67 12.74 12.82 12.89 

7 

3-6 

12.96 13.03 13.10 13.18 13.25 

13.32 13.40 13.47 13.54 13.62 

7 

3-7 

13.69 13.76 13.84 13.91 13.99 

14.06 14.14 14.21 14.29 14.36 

8 

3-8 

14.44 14.52 14.59 14.67 14.75 

14.82 14.90 14.98 15.05 15.13 

8 

3-9 

15.21 15.29 15.37 15.44 15.52 

15.60 15.68 15.76 15.84 15.92 

8 

4.0 

16.00 16.08 16.16 16.24 16.32 

16.40 16.48 16.56 16.65 16.73 

8 

4.1 

16.81 16.89 16.97 17.06 17.14 

17.22 17.31 17.39 I 7-47 I7-56 

8 

4.2 

17.64 17.72 17.81 17.89 17.98 

18.06 18.15 18.23 18.32 18.40 

9 

4-3 

18.49 18.58 18.66 18.75 18.84 

18.92 19.01 19.10 19.18 19.27 

9 

4.4 

19.36 19.45 19.54 19-62 19.71 

19.80 19.89 19.98 20.07 20.16 

9 

4-5 

20.25 20.34 20.43 20.52 20.61 

20.70 20.79 20.88 20.98 21.07 

9 

4.6 

21.16 21.25 21.34 21.44 21.53 

21.62 21.72 21.81 21.90 22.00 

9 

4-7 

22.09 22.18 22.28 22.37 22.47 

22.56 22.66 22.75 22.85 22.94 

10 

4-8 

23.04 23.14 23.23 23.33 23.43 

23.52 23.62 23.72 23.81 23.91 

10 

4.9 

24.01 24.11 24.21 24.30 24.40 

24.50 24.60 24.70 24.80 24.90 

10 

5 -o 

25.00 25.10 25.20 25.30 25.40 

25.50 25.60 25.70 25.81 25.91 

10 

5 -i 

26.01 26.11 26.21 26.32 26.42 

26.52 26.63 26.73 26.83 26.94 

10 

5 2 

27.04 27.14 27.25 27.35 2 7-46 

27.56 27.67 27.77 27.88 27.98 

11 

5-3 

28.09 28.20 28.30 28.41 28.42 

28.62 28.73 28.84 28.94 29.05 

11 

5-4 

29.16 29.27 29.38 29.48 29.59 

29.70 29.81 29.92 30.03 30.14 

11 

n 

01234 

56789 

Diff. 


















Mathematical Tables. Art. 205 


549 


•Table E. Squares of Numbers ( Continued ) 


n 

01234 

56789 

Diff. 

5-5 

30.25 30.36 30.47 30.58 30.69 

30.80 30.91 31.02 31.14 31.25 

11 

5-6 

31.36 31.47 31.58 31.70 31.81 

31.92 32.04 32.15 32.26 32.38 

11 

5-7 

32.49 32.60 32.72 32.83 32.95 

33.06 33.18 33.29 33.41 33.52 

12 

5-8 

33-64 33-76 33.87 33.99 34.11 

34.22 34.34 34.46 34.57 34.69 

12 

5-9 

34.81 34.93 35.05 35.16 35.28 

35-40 35.52 35.64 35-76 35.88 

12 

6.o 

36.00 36.12 36.24 36.36 36.48 

36.60 36.72 36.84 36.97 37.09 

12 

6.i 

37 - 2 i 37.33 37.45 37.58 37.70 

37.82 37.95 38.07 38.19 38.32 

12 

6.2 

38.44 38.56 38.69 38.81 38.94 

39.06 39.19 39.31 39.44 39.56 

13 

6-3 

39.69 39.82 39.94 40.07 40.20 

40.32 40.45 40.58 40.70 40.83 

13 

6.4 

40.96 41.09 41.22 41.34 41-47 

41.60 41.73 41.86 41.99 42.12 

13 

6-5 

42.25 42.38 42.51 42.64 42.77 

42.90 43.03 43.16 43.30 43.43 

13 

6.6 

43.56 43.69 43.82 43.96 44.09 

44.22 44.36 44.49 44.62 44.76 

13 

6.7 

44.89 45.02 45.16 45.29 45.43 

45.56 45.70 45.83 45-97 46.10 

14 

6.8 

46.24 46.38 46.51 46.65 46.79 

46.92 47.06 47.20 47.33 47.47 

14 

6.9 

47.61 47.75 47.89 48.02 48.16 

48.30 48.44 48.58 48.72 48.86 

14 

7.0 

49.00 49.14 49-28 49.42 49.56 

49.70 49.84 49-98 50.13 50.27 

14 

7 -i 

50.41 50.55 50.69 50.84 50.98 

51.12 51.27 51.41 51.55 51.70 

14 

7.2 

51.84 51.98 52.13 52.27 52.42 

52.56 52.71 52.85 53.00 53.14 

1 5 

7-3 

53-29 53-44 53-58 53.73 53.88 

54.02 54.17 54.32 54.46 54.61 

15 

7-4 

54.76 54.91 55.06 55.20 55.35 

55-50 55.65 55.80 55.95 56.10 

15 

7 5 

56.25 56.40 56.55 56.70 56.85 

57.00 57.15 57.30 57-46 57.61 

15 

7.6 

57-76 57.91 58.06 58.22 58.37 

58.52 58.68 58.83 58.98 59.14 

i 5 

7-7 

59.29 59.44 59.60 59.75 59-9 1 

60.06 60.22 60.37 60.53 60.68 

16 

7.8 

60.84 61.00 61.15 61.31 61.47 

61.62 61.78 61.94 62.09 62.25 

16 

7-9 

62.41 62.57 62.73 62.88 63.04 

63.20 63.36 63.52 63.68 63.84 

16 

8 0 

64.00 64.16 64.32 64.48 64.64 

64.8c 64.96 65.12 65.29 65.45 

16 

8.1 

65.61 65.77 65.93 66.10 66.26 

66.42 66.59 66.75 66.91 67.08 

16 

8.2 

67.24 67.40 67.57 67.73 6 7-90 

68.06 68.23 68.39 68.56 68.72 

17 

8-3 

68.89 69.06 69.22 69.39 69.56 

69.72 69.89 70.06 70.22 70.39 

17 

8.4 

70.56 70.73 70.90 71.06 71.23 

71.40 71.57 71.74 7i-9i 72.08 

17 

8-5 

72.25 72.42 72.59 72.76 72.93 

73.10 73.27 73.44 73-62 73-79 

1 7 

8.6 

73.96 74.13 74.3° 74-48 74.65 

. 74.82 75.00 75.17 75-34 75-52 

17 

8-7 

75.69 75.86 76.04 76.21 76.39 

76.56 76.74 76.91 77.09 77-26 

18 

8.8 

77.44 77.62 77.79 77.97 78-15 

78.32 78.50 78.68 78.85 79.03 

18 

8.9 

79.21 79.39 79.57 79-74 79 - 9 2 

80.10 80.28 80.46 80.64 80.82 

18 

9.0 

81.00 81.18 81.36 81.54 81.72 

81.90 82.08 82.26 82.45 82.63 

18 

9 - 1 

82.81 82.99 83.17 83.36 83.54 

83.72 83.91 84.09 84.27 84.46 

18 

9.2 

84.64 84.82 85.01 85.19 85.38 

85-56 85.75 85.93 86.12 86.30 

19 

9-3 

86.49 86.68 86.86 87.05 87.24 

87.42 87.61 87.80 87.98 88.17 

19 

9-4 

88.36 88.55 88.74 88.92 89.11 

89.30 89.49 89.68 89.87 90.06 

19 

9 5 

90.25 90.44 90.63 90.82 91.01 

91.20 9139 9 t -58 9 i -78 9 j -97 

19 

9.6 

92.16 92.35 92.54 92.74 92.93 

93.12 93.32 93.51 93-70 93-90 

19 

9-7 

94.09 94.28 94.48 94-67 94-87 

95.06 95.26 95.45 95-65 95-84 

20 

9.8 

96.04 96.24 96.43 96.63 96.83 

97.02 97.22 97.42 97.61 97.81 

20 

9.9 

98.01 98.21 98.41 98.60 98.80 

99.00 99.20 99.40 99.60 99.00 

20 

n 

01234 

5 6 7 8 9 

Diff. 



















550 


Appendix 


Table F. Areas of Circles 


d 

o i 234 

56789 

Diff. 

1.0 

.7854 .8012 .8171 .8332 .8495 

.8659 .8825 .8992 .9161 .9331 


1.1 

.9503 .9677 .9852 1.003 I-02I 

I.O39 I.057 1.075 1-094 I.” 2 


1.2 

I.I3I I.I50 I.169 I.l88 1.208 

1.227 I.247 I.267 I.287 I.307 

19 

1.3 

I.327 I.348 I.368 I.389 I.4IO 

1.431 1.453 1.474 1496 I * 5 I 7 

21 

1.4 

I.539 I.561 I.584 I.606 I.629 

1.651 1.674 1.697 I * 7 2 ° 1-744 

22 

i -5 

I.767 I.79I I.815 I.839 I.863 

1.887 1.911 1.936 1.961 1.986 

24 

i.6 

2.011 2.036 2.061 2.087 2.112 

2.138 2.164 2.190 2.217 2.243 

26 

i -7 

2.270 2.297 2.324 2.351 2.378 

2.405 2.433 2.461 2.488 2.516 

27 

i.8 

2-545 2.573 2.602 2.63O 2.659 

2.688 2.717 2.746 2.776 2.806 

29 

1.9 

2.835 2.865 2.895 2.926 2.956 

2.986 3.017 3.048 3.079 3.110 

30 

2.0 

3.142 3.I73 3.205 3.237 3.269 

3-301 3.333 3.365 3.398 3.431 

32 

2.1 

3.464 3.497 3-530 3 563 3-597 

3.631 3.664 3.698 3.733 3.767 

34 

2.2 

3.801 3.836 3.871 3.906 3.941 

3.976 4.012 4.047 4.083 4.119 

35 

2-3 

4.155 4.191 4.227 4.264 4.301 

4-337 4-374 4-412 4449 4486 

36 

2.4 

4.524 4.562 4.600 4.638 4.676 

4.714 4.753 4-792 4-831 4-870 

38 

2-5 

4.909 4.948 4.988 5.027 5.067 

5.107 5.147 5.187 5.228 5.269 

40 

2.6 

5-309 5 - 35 ° 5-391 5.433 5.474 

5 . 5 I 5 5-557 5-599 5-641 5-683 

4 i 

2.7 

5.726 5.768 5.811 5.853 5.896 

5.940 5.983 6.026 6.070 6.114 

43 

2.8 

6.158 6.202 6.246 6.290 6.335 

6.379 6.424 6.469 6.514 6.560 

44 

2.9 

6.605 6.651 6.697 6.743 6.789 

6.835 6.881 6.928 6.975 7.022 

46 

3-0 

7.069 7.116 7.163 7.211 7.258 

7.306 7.354 7.402 7.451 7.499 

48 

3 -i 

7.548 7-596 7.645 7.694 7-744 

7.793 7-843 7-892 7-942 7.992 

49 

3-2 

8.042 8.093 8.143 8.194 8.245 

8.296 8.347 8.398 8.450 8.501 

5 i 

3-3 

8.553 8-605 8.657 8.709 8.762 

8.814 8.867 8.920 8.973 9.026 

52 

3-4 

9.079 9.133 9.186 9.240 9.294 

9.348 9.402 9.457 9.511 9.566 

54 

3-5 

9.621 9.676 9.731 9.787 9.842 

9.898 9.954 10.01 10.07 10.12 

56 

3-6 

10.18 10.24 10.29 IO -35 10.41 

10.46 10.52 10.58 10.64 10.69 

6 

3-7 

10.75 10 81 10.87 IO -93 10.99 

11.04 11.10 11.16 11.22 11.28 

6 

3-8 

11.34 11.40 11.46 11.52 11.58 

11.64 11.70 11.76 11.82 11.88 

6 

3-9 

11.95 12.01 12.07 12.13 12.19 

12.25 12.32 12.38 12.44 12.50 

6 

4.0 

12.57 12.63 12.69 12.76 12.82 

12.88 12.95 I 3-° I I 3-°7 13-14 

7 

4.1 

13.20 13.27 13.33 13-40 1346 

13-53 13-59 i 3 - 66 I 3 - 7 2 13-79 

/ 

4.2 

13.85 13.92 13.99 14.05 14.12 

14.19 14.25 14.32 14.39 14.45 

7 

4-3 

14.52 14.59 14.66 14.73 14-79 

14.86 14.93 15-00 15.07 15.14 

/ 

4-4 

15.21 15.27 15.34 15.41 15.48 

15.55 15-62 15.69 15.76 15.83 

7 

4-5 

15.90 15.98 16.05 16.12 16.19 

16.26 16.33 16.40 16.47 16.55 

7 

4.6 

16.62 16.69 16.76 16.84 16.91 

16.98 17.06 17.13 17.20 17.28 

7 

4-7 

17.35 1742 17-50 17-57 17-65 

17.72 17.80 17.87 17.95 18.02 

8 

4.8 

18.10 18.17 18.25 18.32 18.40 

18.47 18.55 18.63 18.70 18.78 

8 

4-9 

18.86 18.93 19.01 19.09 19.17 

19.24 19.32 19.40 19.48 19.56 

8 

5 -o 

19.63 19.71 19.79 19.87 19.95 

20.03 20.11 20.19 20.27 20.35 

8 

5 -i 

20.43 20.51 20.59 20.67 20.75 

20.83 20.91 20.99 21.07 21.16 

8 

5-2 

21.24 21.32 21.40 21.48 21.57 

21.65 21.73 21.81 21.90 21.98 

8 

5-3 

22.06 22.15 22.23 22.31 22.40 

22.48 22.56 22.65 22.73 22.82 

8 

5-4 

22.90 22.99 23.07 23.16 23.24 

23.33 23.41 23.50 23.59 23.67 

9 

d 

01 234 

56789 

Diff. 


















Mathematical Tables. 


Art. 205 


551 


Table F. Areas of Circles ( Continued ) 


d 

01234 

56789 

Diff. 

5-5 

23.76 23.84 23.93 24-02 24.11 

24.19 24.28 24.37 24.45 24-54 

9 

5-6 

24.63 24.72 24.81 24.89 24.98 

25.07 25.16 25.25 25.34 25.43 

9 

5-7 

25.52 25.61 25.70 25.79 25.88 

25.97 26.06 26.15 26.24 26.33 

9 

5-8 

26.42 26.51 26.60 26.69 26.79 

26.88 26.97 27.06 27.15 27.25 

9 

5-9 

27.34 27.43 27.53 27.62 27.71 

27.81 27.90 27.99 28.09 28.18 

9 

6.0 

28.27 28.37 28.46 28.56 28.65 

28.75 28.84 28.94 29.03 29.13 

9 

6.1 

29.22 29.32 29.42 29.51 29.61 

29.71 29.80 29.90 30.00 30.09 

10 

6.2 

30.19 30.29 30.39 30.48 30.58 

30.68 30.78 30.88 30.97 31.07 

10 

6-3 

31.17 31.27 31.37 31.47 31.57 

31.67 31.77 31.87 31.97 32.07 

10 

6.4 

32.17 32.27 32.37 32.47 32.57 

32.67 32.78 32.88 32.98 33.08 

10 

6-5 

33.18 33.29 33.39 33-49 33-59 

33-70 33.80 33.90 34.00 34.11 

10 

6.6 

34.21 34.32 34.42 34.52 34.63 

34.73 34.84 34-94 35-05 35-15 

10 

6.7 

35-26 35-36 35.47 35-57 35-68 

35.78 35.89 36.00 36.10 36.21 

10 

6.8 

36.32 36.42 36.53 36.64 36.75 

36.85 36.96 37.07 37.18 37.28 

11 

6.9 

37-39 37 - 5 ° 37-6i 37.72 37.83 

37.94 38.05 38.16 38.26 38.37 

11 

7.0 

38.48 38.59 38.70 38.82 38.93 

39.04 39.15 39.26 39.37 39.48 

11 

7 -i 

39-59 39-70 39.82 39.93 40.04 

40.15 40.26 40.38 40.49 40.60 

11 

7.2 

40.72 40.83 40.94 41.06 41.17 

41.28 41.40 41.51 41.62 41.74 

11 

7-3 

41.85 41.97 42.08 42.20 42.31 

42.43 42.54 42.66 42.78 42.89 

11 

7-4 

43.01 43.12 43.24 43.36 43.47 

43-59 43-71 43-83 43-94 44-o6 

12 

7-5 

44.18 44.30 44.41 44.53 44.65 

44.77 44.89 45 - 0 ! 45-13 45- 2 5 

12 

7.6 

45.36 45-48 45-6o 45-72 45-84 

45.96 46.08 46.20 46.32 46.45 

12 

7-7 

46.57 46.69 46.81 46.93 47.05 

47.17 47.29 47.42 47.54 47.66 

12 

7.8 

47.78 47.91 48.03 48.15 48.27 

48.40 48.52 48.65 48.77 48-89 

12 

7-9 

49.02 49.14 49-27 49-39 49 - 5 1 

49.64 49.76 49.89 50.01 50.14 

12 

8.0 

50.27 50.39 50.52 50.64 50.77 

50.90 51.02 51.15 51.28 51.40 

13 

8.1 

51-53 5i-66 51.78 51.91 52.04 

52.17 52.30 52.42 52.55 52.68 

13 

8.2 

52.81 52.94 53-07 53-20 53.33 

53-4^ 53.59 53.72 53.85 53.98 

13 

8-3 

54.11 54.24 54.37 54-50 54-63 

54.76 54.89 55.02 55.15 55.29 

13 

8.4 

55-42 55.55 55-68 55.81 55.95 

56.08 56.21 56.35 56.48 56.61 

13 

8-5 

56.75 56.88 57.01 57.15 57-28 

57.41 57.55 57-68 57.82 57.95 

13 

8.6 

58.09 58.22 58.36 58.49 58.63 

58.77 58.90 59.04 59.17 59 - 3 i 

14 

8.7 

59-45 59-58 59.72 59.86 59.99 

60.13 60.27 60.41 60.55 60.68 

14 

8.8 

60.82 60.96 61.10 61.24 61.38 

61.51 61.65 61.79 61.93 62.07 

14 

8.9 

62.21 62.35 62.49 62.63 62.77 

62.91 63.05 63.19 63.33 63.48 

14 

9 -° 

63.62 63.76 63.90 64.04 64.18 

64.33 64-47 64.61 64.75 64.90 

14 

9.1 

65.04 65.18 65.33 65.47 65.61 

6^.76 6^.90 66.04 66.19 66.33 

14 

9.2 

66.48 66.62 66.77 66.91 67.06 

67.20 67.35 67.49 67.64 67.78 

15 

9-3 

67.93 68.08 68.22 68.37 68.51 

68.66 68.81 68.96 69.10 69.25 

15 

9.4 

69.40 69.55 69.69 69.84 69.99 

70.14 70.29 70.44 70-58 7°-73 

15 

9-5 

70.88 71.03 71.18 71.33 7^48 

71.63 71-78 7i-93 72.o8 72.23 

15 

9.6 

72.38 72.53 72.68 72.84 72.99 

73.14 73.29 73-44 73-59 73-75 

15 

9-7 

73.90 74.05 74.20 74.36 74.51 

74.66 74.82 74.97 75 - 12 75- 2 8 

15 

9.8 

75-43 75-58 75-74 75-89 76.05 

76.20 76.36 76.51 76.67 76.82 

ID 

- c. 

9.9 

76.98 77.13 77-29 77-44 77-6o 

77.76 77.91 78.07 78.23 78.38 

10 

d 

01234 

56789 

Diff. 
























552 


Appendix 


Table G. Trigonometric Functions 


Angle 

Deg. 

Arc 

Sin 

Tan 

Sec 

Cosec 

Cot 

Cos 

Coarc 


O 

0. 

O. 

O. 

I. 

00 

00 

1. 

1.5708 

90 

I 

0.0175 

O.OI75 

O.0175 

1.0002 

57-299 

57.290 

0.9998 

•5533 

89 

2 

•0349 

•0349 

•0349 

1.0006 

28.654 

28.636 

•9994 

•5359 

88 

3 

.0524 

.0523 

.0524 

1.0014 

19.107 

19.081 

.9986 

• 5 i 84 

87 

4 

.0698 

.0698 

.0699 

I.OO24 

14.336 

14.301 

.9976 

.5010 

86 

5 

.0873 

.0872 

•0875 

I.OO38 

11.474 

11.430 

.9962 

•4835 

85 

6 

0.1047 

O.IO45 

0.1051 

1-0055 

9.5668 

9-5144 

0-9945 

1.4661 

84 

7 

.1222 

.1219 

.1228 

I.OO75 

8.2055 

8-1443 

•9925 

.4486 

83 

8 

.1396 

.1392 

.1405 

I.OO98 

7-1853 

7 -H 54 

•9903 

.4312 

82 

9 

•1571 

.1564 

.1584 

I-OI25 

6.3925 

6.3138 

•9877 

•4137 

81 

IO 

•1745 

.1736 

.1763 

I.OI54 

5-7588 

5 - 67 I 3 

.9848 

•3963 

80 

ii 

0.1920 

.01908 

0.1944 

I.O187 

5.2408 

5.1446 

0.9816 

1.3788 

79 

12 

.2094 

.2079 

.2126 

1.0223 

4.8097 

4.7046 

. 978 i 

.3614 

78 

13 

.2269 

.2250 

.2309 

I.O263 

4-4454 

4-3315 

•9744 

•3439 

77 

14 

.2443 

.2419 

•2493 

I.0306 

4 -I 336 

4.0108 

•9703 

•3265 

76 

15 

.2618 

.2588 

.2679 

1-0353 

3-8637 

3-7321 

•9659 

•3090 

75 

16 

O.2793 

0.2756 

0.2867 

1.0403 

3.6280 

3-4874 

0.9613 

1-2915 

74 

i 7 

.2967 

.2924 

•3057 

1-0457 

3-4203 

3.2709 

•9563 

.2741 

73 

18 

.3142 

.3090 

•3249 

1-0515 

3.2361 

3.0777 

•9511 

.2566 

72 

19 

•3316 

•3256 

•3443 

1.0576 

3.0716 

2.9042 

•9455 

.2392 

7 i 

20 

•3491 

•3420 

.3640 

1.0642 

2.9238 

2-7475 

•9397 

.2217 

70 

21 

0.3665 

O.3584 

0.3839 

1.0711 

2.7904 

2.6051 

0.9336 

1.2043 

69 

22 

.3840 

•3746 

.4040 

1.0785 

2.6695 

2.4751 

.9272 

.1868 

68 

2 3 

.4014 

•3907 

.4245 

1.0864 

2-5593 

2-3559 

.9205 

.1694 

67 

24 

.4189 

.4067 

•4452 

1.0946 

2.4586 

2.2460 

•9135 

.1519 

66 

25 

•4363 

.4226 

.4663 

1.1034 

2.3662 

2.1445 

•9063 

•1345 

65 

26 

0.4538 

0.4384 

0.4877, 

1.1126 

2.2812 

2.0503 

0.8988 

1.1170 

64 

27 

.4712 

•4540 

•5095 

1.1223 

2.2027 

1.9626 

.8910 

.0996 

63 

28 

.4887 

.4695 

•5317 

1.1326 

2.1301 

1.8807 

.8829 

.0821 

62 

29 

.5061 

.4848 

•5543 

I-I 434 

2.0627 

1.8040 

.8746 

.0647 

61 

30 

•5236 

.5000 

•5774 

I-I 547 

2.0000 

1-7321 

.8660 

.0472 

60 

31 

0.5411 

0.5150 

0.6009 

1.1666 

1.9416 

1.6643 

0.8572 

1.0297 

59 

32 

•5585 

•5299 

.6249 

1.1792 

1.8871 

1.6003 

.8480 

1.0123 

58 

33 

.5760 

.5446 

.6494 

1.1924 

1.8361 

1-5399 

•8387 

0.9948 

57 

34 

•5934 

•5592 

•6745 

1.2062 

1.7883 

1.4826 

.8290 

•9774 

56 

35 

.6109 

•5736 

.7002 

1.2208 

1-7434 

1.4281 

.8192 

•9599 

55 

36 

0.6283 

0.5878 

0.7265 

1.2361 

1-7013 

I -3764 

0.8089 

0.9425 

54 

37 

.6458 

.6018 

•7536 

1-2521 

1.6616 

1.3270 

.7986 

.9250 

53 

38 

.6632 

.6157 

•7813 

1.2690 

1.6243 

1.2799 

.7880 

.9076 

52 

39 

.6807 

.6293 

.8098 

1.2868 

1.5890 

1-2349 

.7771 

.8901 

51 

40 

.6981 

.6428 

.8391 

I -3054 

1-5557 

1.1918 

.7660 

•8727 

50 

4 i 

0.7156 

0.6561 

0.8693 

1-3250 

1-5243 

1.1504 

0-7547 

0.8552 

49 

42 

•7330 

.6691 

.9004 

1-3456 

1-4945 

1.1106 

•7431 

•8378 

48 

43 

•7505 

.6820 

•9325 

I -3673 

1.4663 

1.0724 

•7314 

.8203 

47 

44 

.7679 

.6947 

•9657 

1.3902 

1.4396 

1-0355 

•7193 

.8029 

46 

45 

•7854 

.7071 

1. 

1.4142 

1.4142 

1. 

.7071 

•7854 

45 


Coarc 

Cos 

Cot 

Cosec 

Sec 

Tan 

Sin 

Arc 

Angle 

Deg. 












































Mathematical Tables. Art. 205 


553 


Table H. Logarithms of Trigonometric Functions 


Angle 

Deg. 

Log Arc 

Log Sin 

Log Tan 

Log Sec 

Log 

Cosec 

Log Cot 

Log Cos 

Log 

Coarc 


o 

— CO 

— 00 

— 00 

O. 

00 

00 

0. 

0.1961 

90 

I 

2.2419 

2.2419 

2.2419 

0.0001 

1.7581 

1.7581 

1.9999 

•1913 

89 

2 

•5429 

.5428 

.5431 

.0003 

•4572 

.4569 

•9997 

.1864 

88 

3 

.7190 

.7188 

.7194 

.0006 

.2812 

.2806 

•9994 

.1814 

87 

4 

•8439 

.8436 

.8446 

.0011 

.1564 

.1554 

.9989 

.1764 

86 

5 

.9408 

.9403 

.9420 

.0017 

.0597 

.0580 

•9983 

•1713 

85 

6 

1.0200 

I .OI92 

1.0216 

0.0024 

0.9808 

0.9784 

1.9976 

0.1662 

84 

7 

.0870 

.0859 

.0891 

.0032 

.9141 

.9109 

.9968 

.1610 

83 

S 

.1450 

.1436 

.1478 

.0042 

.8564 

.8522 

.9958 

•1557 

82 

9 

.1961 

.1943 

.1997 

.0054 

.8057 

.8003 

.9946 

.1504 

81 

IO 

.2419 

.2397 

.2463 

.0066 

.7603 

•7537 

•9934 

.1450 

80 

11 

1.2833 

1.2806 

1.2887 

0.0081 

0.7194 

O.7113 

T . 99 i 9 

0.1395 

79 

12 

.3211 

•3 1 79 

•3275 

.0096 

.6821 

.6725 

.9904 

.1340 

78 

13 

.3558 

.3521 

•3634 

.0113 

.6479 

.6366 

.9887 

.1284 

77 

i 4 

.3880 

•3837 

.3968 

.0131 

.6163 

.6032 

.9869 

.1227 

76 

i 5 

.4180 

.4130 

.4281 

.0151 

.5870 

•5719 

•9849 

.1169 

75 

16 

1.4460 

1.4403 

1-4575 

O.OI 72 

0-5597 

0.5425 

1.9828 

O.IIII 

74 

i 7 

.4723 

.4659 

•4853 

.0194 

■5341 

• 5 I 47 

.9806 

.1052 

73 

18 

.4971 

.4900 

.5118 

.0218 

.5100 

.4882 

.9782 

.0992 

72 

i9 

.5206 

.5126 

•5370 

.0243 

•4874 

.4630 

•9757 

.0931 

7i 

20 

•5429 

•5341 

.5611 

.0270 

.4659 

.4389 

•9730 

.0870 

70 

21 

1.5641 

1-5543 

1.5842 

0.0298 

0-4457 

0.4158 

1.9702 

0.0807 

69 

22 

.5843 

•5736 

.6064 

.0328 

.4264 

•3936 

.9672 

•0744 

68 

23 

.6036 

.5919 

.6279 

.0360 

.4081 

.3721 

.9640 

.0680 

67 

24 

.6221 

.6093 

.6486 

.0393 

•3907 

• 35 M 

.9607 

.0614 

66 

25 

.6398 

.6259 

.6687 

.0427 

.3741 

.3313 

•9573 

.0548 

65 

26 

1.6569 

1.6418 

1.6882 

0.0463 

0.3582 

0.3118 

1-9537 

0.0481 

64 

27 

.6732 

.6570 

.7072 

.0501 

•3430 

.2928 

•9499 

.0412 

63 

28 

.6890 

.6716 

•7257 

.0541 

.3284 

•2743 

•9459 

•0343 

62 

29 

•7043 

.6856 

•7438 

.0582 

•3144 

.2562 

.9418 

.0272 

61 

30 

.7190 

.6990 

.7614 

.0625 

• 3 010 

.2386 

•9375 

.0200 

60 

31 

1-7332 

1.7118 

1.7788 

0.0669 

0.2882 

0.2212 

L9331 

O.OI27 

59 

32 

.7470 

.7242 

•7958 

.0716 

.2758 

.2042 

.9284 

O.OO53 

58 

33 

.7604 

.7361 

.8125 

.0764 

.2639 

.1875 

.9236 

1.9978 

57 

34 

•7734 

.7476 

.8290 

.0814 

.2524 

.1710 

.9186 

.9901 

56 

35 

•7859 

.7586 

.8452 

.0866 

.2414 

.1548 

•9134 

.9822 

55 

36 

Y.7982 

1.7692 

1.8613 

0.0920 

0.2308 

0.1387 

1.9080 

1-9743 

54 

37 

.8101 

•7795 

.8771 

.0977 

.2205 

.1229 

.9023 

.9662 

53 

38 

.8217 

•7893 

.8928 

•1035 

.2107 

.1072 

.8965 

•9579 

52 

39 

.8329 

.7989 

.9084 

.1095 

.2011 

.0916 

•8905 

•9494 

5 i 

40 

.8439 

.8081 

•9238 

.1157 

.1919 

.0762 

.8843 

.9408 

50 

41 

1.8547 

1.8169 

1.9392 

0.1222 

0.1831 

0.0608 

1.8778 

1.9321 

49 

42 

.8651 

.8255 

•9544 

.1289 

• 1745 

.0456 

.8711 

.9231 

48 

43 

.8753 

.8338 

.9697 

•1359 

.1662 

.0303 

.8641 

.9140 

47 

44 

.8853 

.8418 

.9848 

• I 43 I 

.1582 

.0152 

.8569 

.9046 

46 

45 

.8951 

.8495 

0. 

.1505 

.1505 

O. 

.8495 

.8951 

45 


Log 

Coarc 

Log Cos 

Log Cot 

Log 

Cosec 

Log Sec 

Log Tan 

Log Sin 

Log Arc 

Angle 

Deg. 








































554 


Appendix 


Table J. Logarithms of Numbers 


n 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

IO 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

42 

ii 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

38 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

35 

13 

ii 39 

ii 73 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

32 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

30 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

28 

l6 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

27 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

25 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

24 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

22 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

20 

22 

3424 

3444 

3464 

3483 

3502 

3522 

354 i 

356 o 

3579 

3598 

19 

23 

3617 

3636 

3655 

3674 

3692 

37 ii 

37 29 

3747 

3766 

3784 

18 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

18 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

i 7 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

17 

27 

43 U 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

16 

28 

4472 

4487 

4502 

45 i 8 

4533 

4548 

4564 

4579 

4594 

4609 

15 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

15 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

14 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

14 

32 

5051 

5065 

5079 

5092 

5105 

5 ii 9 

5132 

5145 

5159 

5172 

J 3 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

13 

34 

5315 

5328 

5340 

5353 

5366 

5378 

539 i 

5403 

54 i 6 

5428 

13 

35 

544 i 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

555 i 

1 ? 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

12 

37 • 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

12 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

11 

39 

59 ii 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

11 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

11 

4 i 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

11 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

10 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

10 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

10 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

10 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

9 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

9 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

9 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

9 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

9 

5i 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

8 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

72 35 

8 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

8 

54 

7324 

7332 

7340 

7348 

7356 

7364 

737 2 

7380 

7388 

7396 

8 

n 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 
























Mathematical Tables. Art. 205 


555 


Table J. Logarithms of Numbers ( Continued ) 


n 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

55 

7404 

7412 

7419 

7427 

7435 

7443 

745i 

7459 

7466 

7474 

8 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

755i 


57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 


58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 


59 

7709 

7716 

7723 

773i 

7738 

7745 

7752 

7760 

7767 

7774 


60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

7 

Oi 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

79 x 7 


62 

7924 

793i 

7938 

7945 

7952 

7959 

7966 

7973 

798o 

7987 


63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 


64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 


65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

7 

66 

8i95 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 


67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 


68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 


69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 


70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

6 

7 r 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 


7 2 

8573 

8579 

8485 

8591 

8597 

8603 

8609 

8615 

8621 

8627 


73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 


74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739 

8745 


75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

6 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 


77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 


78 

8021 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 


79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 


80 

9031 

9036 

9042 

9°47 

9053 

9058 

9063 

9069 

9074 

9079 

5 i 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9 n 7 

9122 

9128 

9i33 


82 

9138 

9143 

9 T 49 

9154 

9159 

9165 

9170 

9175 

9180 

9186 


83 

919 1 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 


84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 


85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

5 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 


87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

• 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

I i 

89 

9494 

9499 

9504 

9509 

9513 

95i8 

9523 

9528 

9533 

9538 


90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

958 i 

9586 

5 

91 

959° 

9595 

9600 

9605 

9609 

| 9614 

9619 

9624 

9628 

9633 


Q2 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 


93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 


94 

973 1 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 


95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

4 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 


97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 


98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 


99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 


n 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Diff. 

1 


























556 


Appendix 


Table K. Constants and their Logarithms 


Name 

(Radius of circle or sphere = i) 

Symbol 

Number 

Logarithm 

Area of circle 

7r 

3.141 592 654 

O.497 149 873 

Circumference of circle 

2 7 T 

6.283 185 307 

0.798 179 868 

Surface of sphere 

477- 

12.566 370 614 

1.099 209 864 


l T 

0-523 598 776 

I.718 998 622 

Quadrant of circle 


0.785 398 163 

1.895 089 881 

Area of semicircle 


1.570 796327 

0.196 119 877 

Volume of sphere 

1 T 

4.187 790 205 

0.622 088 609 


7t2 

9.869 604 401 

0.994 299 745 


7 T& 

1.772 453 851 

0.248 574 936 

Degrees in a radian 

i8o/7t 

57.295 779 513 

1.758 122 632 

Minutes in a radian 

10800/7r 

3 437-746 77 i 

3.536 273 883 

Seconds in a radian 

648000/ TT 

206 264.806 

5.314425 133 


I / 7 T 

0.318 309 886 

I.502 850 127 


1/7 

0.564 189 584 

1.751 425 064 


i/tt 2 

0.101 321 184 

1.005 700 255 

Circumference / 360 

arc i° 

0.017 453 293 

2.241 877 368 


• O 

sin 1 

0.017 452 406 

2.241 855 318 

Circumference/ 21600 

arc 1' 

0.000 290 888 

4.463 726 117 


sin i' 

0.000 290 888 

4.463 726 hi 

Circumference/1296000 

arc 1" 

0.000 004 848 

6.685 574 867 


sin 1 " 

0.000 004 848 

6.685 574 867 

Base Naperian system of logs 

e 

2.718 281 828 

0.434 294 482 

Modulus common system of logs 

M 

0.434 294 482 

1.637 784311 

Naperian log of 10 

i/M 

2.302 585 093 

0.362 215 689 


hr 

0.476 936 3 

1.678 460 4 

Probable error constant 

hr V 2 

0.674 489 7 

1.828975 4 
















Index 


557 


INDEX 


(The numbers refer to pages.) 


Absolute velocity, 60, 64, 422, 440 
Acceleration, 3, n, 12, 21, 546 
Acre-foot, 375 
Adjutage, 178, 191 
Advantageous angle, 420 
nozzle, 449 
section, 283 

velocity, 421, 436, 448, 469, 472, 
482 

Air chamber, 242, 424, 510 
Air-lift pump, 528 
Air valve, 224, 248 
Anchor ice, 5 
Angle measurements, 108 
Answers to problems, 544 
Approach, angle of, 236, 445 
velocity of, 51, 123, 145-153 
apron of dam, 163 
Aqueducts, 210, 272, 300 
Archimedean screw, 504 
Areas of circles, 545, 556 
Atmospheric pressure, 2, 7, 20, 26, 41, 
188, 472, 507 
Automatic devices, 251 

Backpitch wheel, 450 
Backwater, 344, 353, 355 
function, 354 
Ball nozzle, 199 
Barker’s mill, 453 
Barometer, 7, 8, 20, 472, 507 
Bazin’s formula, 298, 316 
Bends in rivers, 411 
Bernouilli’s theorem, 68, 203 
Blow-offs, 224 
Boiling point, 8, 20 
Bore, 350, 352 
Bridge piers, 342 


Bristol water level gage, 76 
Boyden diffuser, 476 
hook gage, 79 
turbine, 395, 462 
Brake, friction, 389 
Branched pipes, 254 
hose, 534 

Breast wheels, 437, 528 
Brick conduits, 295, 206 
sewers, 292 
Brooks, 272, 317 
Buckets, 435, 437, 450, 505 
Bucket pumps, 13 
Buoyancy, center of, 30 

Canal boat, 490 
lock, 136 
Canals, 272-292 
Cascade wheel, 441 
Cast-iron pipes, 258, 295 
Catskill aqueduct, 300, 336 
Center of buoyancy, 30, 499 
of gravity, 31 
of pressure, 34, 36 
Centrifugal force, 62 
pump, 521 

Chain pump, 13, 528 
Channels, 272-317 
Chemical methods for velocity, 334 
Chezy’s formula, 275, 287, 313, 315 
Cippoletti weir, 170 
Circles, areas of, 545, 550 
properties of, 280, 556 
Circular conduit, 276, 279, 280 
orifices, 46, 116, 138 
Classification of pumps, 505, 527 
of surfaces, 295, 304 
of turbines, 447 




558 

Coal used by steamers, 490 
Cock valve, 223 
Coefficient of contraction, in 
nozzles, 189 
orifices, 112, 129 
tubes, 184, 185 
Coefficient of discharge, 115 
channels, 293, 313 
dams, 176 
nozzles, 189 

orifices, 118, 119, 121, 123 
pipes, 201, 297, 298 
sewers, 292 

tubes, 185, 189, 192, 195 
turbines, 456 
weirs, 150, 152, 174, 175 
Coefficient of roughness, 289, 297 
Coefficient of velocity, 113 
nozzles, 189 
orifices, 114 
tubes, 185, 195 
Compound pipes, 240, 543 
tubes, 191 

Compressed air, 530 
Compressibility of water, 5, 20 
Computations, 15-22, 72, 138 
Conduit pipes, 295 
Conduits, 272-317 
Conical tubes, 189 
wheel, 451 

Conservation of energy, 47, 193 
Constants, tables of, 546, 556 
Consumption of water, 376 
Contracted weirs, 141, 149, 174 
Contraction, of a jet, no 
coefficient of, in 
gradual, 182 
sudden, 181 
suppression of, 127 
Converging tubes, 191 
Cotton hose, 264 
Crest, of a weir, 80, 142, 160 
of a dam, 342 
rounded and wide, 160 
Critical velocity, 269 
Cross-section, velocities in, 320 
Croton aqueduct, 300, 301 
Cubic feet, 2, 546 


Index 

Current indicators, 325 
meters, 96, 324, 336 
Curvature factors, 218 
Curved surfaces, 31 
Curves, backwater, 161, 343 
in pipes, 238, 245, 409 
in rivers, 409 
Cuttlefish, 493 
Cutwater of piers, 344 
Cycle of rainfall, 378 

Dams, 39, 40, 43, 162, 176, 342 
Danalde, 451 
Data, fundamental, 1-22 
Depth of flotation, 28 
Design of turbines, 469 
of power plants, 364 
of water wheels, 451 
Diameters of pipes, 230 
water mains, 258, 260, 260 
Differential pressure gages, 85 
Diffuser, 474 
Discharge, 65, 94, 115 
conduits, 272-317 
curves, 331, 339 
fountain flow, 209 
gaging of, 327 
nozzles, 242, 265 
orifices, 109-140 
pipes, 211-271 
rivers, 318-364 
theoretic, 65 
tubes, 177-210 
turbines, 462 
weirs, 141-176, 159 
Discharge curves, 331, 339 
Discharging capacity, 233 
Disk valve, 223 
Displacement pumps, 527 
Distilled water, 6, 19 
Ditches, 272, 292 
Diverging tubes, 191 
Diversions, 254 
Double-acting pump, 512 
Double floats, 322, 336 
Downward-flow wheels, 446 
turbines, 471 



Index 


559 


Draft tube, 460 
Drag of a ship, 489 
Drop-down curve, 360 
function, 361 
Dropping head, 135 
Duplex pump, 513 
Duty of pumps, 518 
water, 375 

Dynamic pressure, 59, 399-431, 486 
Dynamo, 396, 481 
Dynamometer, 381 

Effective head, 53, 124, 386 
power, 388 
Efficiency, 57, 382 
jet, 134 

jet propeller, 493 
motors, 384, 391, 432 
moving vanes, 420 
paddle wheels, 495 
pumps, 504-538 
reaction wheel, 438 
screw propeller, 495 
turbines, 454, 456, 466, 474 
water wheels, 436, 438, 439 
Egg-shaped sewers, 289 
Ejector pump, 529, 530 
Elasticity of water, 10, 20 
Electric analogies, 257, 539 
generators, 396, 385 
Elevations by barometer, 8 
Elliptical orifices, no 
Emptying a canal lock, 137 
a vessel, 69 
End contractions, 149 
Energy, 3, 68, 178 
loss of, 133 
in channels, 312 
tubes, 178, 200 
of a jet, 56 

Engine, hydraulic, 526 
pressure, 528 
pumping, 517 
English measures, 1, 547 
Enlargement of section, 180, 309 
Entrance angle, 446, 466 
Eosine, 334 


Erosion, 294, 341 
Errors in computations, 15, 105 
in measurements, 130, 142 
Eureka turbine, 460 
Evaporation, 369 
Exit angle, 464, 467 
Expansion of section, 179 

Fair form of boat, 486 
Fall increaser, 477 
Falling bodies, n, 44 
Feet and inches, 1 
Filaments, 274 
Filling canal lock, 137 
Filter bed, 249, 250, 268 
Fire hose, 264, 270 
engine, 537 
service, 254 
Floats, 250, 322 
Flotation, depth of, 28 
stability of, 29, 497 
Flow, dynamic pressure, 58 
blood in veins, 268 
canals and conduits, 272-317 
dams, 163-167, 176 
fountain, 208 
jets, 54, 56, 198 
non-uniform, 346 
orifices, 46, 109-140 
pipes, 67, 211-271 
revolving vessel, 62 
rivers, 318-364 
steady, 31, 67 
tubes, 177-210 
turbines, 461, 453-484 
under pressure, 49 
Flume, testing, 396 
Foot, 1, 547 
Foot valve, 509, 513 
Force pump, 7, 505, 510 
Force, unit of, 2 
Forebay, 308, 362, 383 
Foss’ formula, 304 
Fountain flow, 207, 208 
Fourneyron turbine, 456, 476 
Francis turbine, 456 
float formula, 323 





Index 


560 

Francis weir formula, 154 
Free surface, 4, 25 
Frictional resistances, 44 
channels, 273, 295 
pipes, 214 
pumps, 507, 513 
turbines, 432, 458 
water wheels, 403, 434 
ships, 486 
Friction brake, 389 
factors, 259, 261, 270 
heads, 216, 218 
Friez recording gage, 76 


Hammer in water pipes, 412 
Head, 25, 81, 134, 142, 178, 388 
and pressure, 25, 26, 41, 51 
effective, 53 

losses of, 133, 217, 218, 250, 306 
measurement of, 76, 79, 130, 234 
Heat units, 518 
Historical notes, n, 23, 206 
Holyoke tests, 394 
Hook gage, 79, 319, 384 
Horizontal impulse wheels, 444 
range of a jet, 54, 199 


Horse-power, 3, 18, 547 
effective, 388 
nominal, 397 
Horseshoe conduits, 306 
Hose, 264, 270, 534 
House-service pipes, 245 
Hunt turbine, 459 
Hurdy-gurdy wheel, 443 
Hydraulic constants, 546 
engine, 526 
gradient, 237, 239 
jump, 349 
mean depth, 273 
motors, 388, 432-484 
press, 84 
radius, 272, 543 
ram, 524 

Hydraulic-electric analogies, 539 
Hydraulics, defined, 13 
theoretical, 44-74 
Hydromechanics, 13, 416, 486 
Hydrometric balance, 325 
pendulum, 324 
Hydrostatic head, 25, 41, 68 
Hydrostatics, 13, 22-45 

Ice, 4, 5 , 7 , 18, 19 
Immersed bodies, 36, 407 
Impact, 178, 180, 401, 446 
Impeller pump, 528 
Impulse, 58, 399, 401, 408 
turbines, 457, 476 
wheels, 441-450 
Inch, 1, 547 
Inclined pipes, 203 
Inclined tubes, 202 
Incrustations in pipes, 259 
Inertia, moments of, 37, 499 
Injector pump, 528 
Instruments, 75-108 
Inward-flow turbines, 456, 472 
Inward-projecting tubes, 190 
Irrigation, hydraulics, 375 

Jersey City aqueduct, 302 
Jet propeller, 492 


Gages, 2, 75, 76, 79, 81-86, 250, 338, 386 
Gaging flow, 95, 129, 142 
of rivers, 321, 332, 335, 374 
Gallon, 1, 2, 546 
Gate of a turbine, 456, 458, 479 
Gates, pressure on, 38 
Gate valve, 224 
Girard turbine, 476 
Glacier, flow of, 305 
Governor, 483 

Gradient, hydraulic, 237, 239 
Graphic methods, 105 
Gravity, acceleration of, 11, 12, 21, 
44, 485, 546 
center of, 32 
water supply, 377 
Greek letters, 17 
Ground water, 372 
Guides, 469 



Index 


561 


Jet pump, 528 

Jets, 54-60, 196, 205, 404, 442 
contraction of, 2, no 
energy of, 56 
from nozzles, 102, 196 
height of, 199, 209 
impulse of, 56, 58, 418 
on vanes, 417 
path of, 54, 56, 58 
range of, 55, 56 
Jonval turbine, 456 
Jump, 350 

Keely motor, 24 
Kilowatt, 396, 547 
Kinetic energy, 3, 45 
Knot, 485 

Kutter’s formula, 287, 313-316, 319 

Lampe’s formula, 268, 270 
Leakage, 384, 437, 509 
Least squares, method of, 107 
Leffel turbine, 459 
Lift pump, 505 
Lighthouses, 419 
Linen hose, 264 
Liter, 547 
Lock-bar pipe, 262 
Lock of canal, 136 
Log, nautical, 323, 485 
Logarithms, 15, 553 ~ 55<5 
Long pipes, 230 
tubes, 200 

Loss of head, 133, 217, 218, 250, 306 
contraction, 181, 182 
curvature, 218, 222 
entrance, 213 
expansion, 186 
friction, 194, 212, 214 
Loss of weight in water, 27 
Lowell tests, 394 

Masonry dams, 40, 43 
conduits, 300 

Mathematical tables, 545-556 


Mean velocity, 92, 225, 274, 275, 322, 330 
Measurement of water, 77, 129, 384 
Measuring instruments, 75-108 
Mercury, 7, 51, 83, 84 
Mercury gage, 83, 85 
Metacenter, 30, 498 
Meter, 547 

Meters, current, 96, 324 
Premier, 93 
Simplex, 92 
Venturi, 89 
water, 88, 132 

Method of least squares, 107 
Metric measures, 3, 18, 41, 72, 138, 173, 
210, 269, 312, 547 
Mile, 485 
Mill power, 396 
Miner's inch, 131 
Mississippi river, 321 
Module, 132 

Modulus of elasticity, 10, 20, 414 
Moments of inertia, 37, 499 
Motors, hydraulic, 386, 391 
Mouthpiece, 191 
Moving vanes, 419 
Mud valves, 224 

Nautical mile, 485 
Naval hydromechanics, 485 
Navigation canals, 362 
Negative pressure, 69 
Niagara power plants, 394, 478 
turbines, 477 
Non-uniform flow, 346 
Normal pressure, 31 

Nozzles, 102, 196, 242, 387, 442, 448, 

529 

jets from, 102, 199, 219 
Numerical computations, 15 

Oar, action of, 494 
Oblique weirs, 172 
Observations, discussion of, 75-108 
Obstructions in channels, 302 
in pipes, 259 

Ocean waves, 351, 408, 501 




562 


Index 


Ogee dams, 165 
Ohm’s law, 539 
Oil, 51, 86 

Oil differential gage, 87 
Operating devices, 248 
Orifices, 46, 109-140, 387 
Oscillations, 497, 543 
Outward-flow turbine, 444 
Overshot wheels, 434, 528 

Paddle wheels, 493 
Paraboloid, 63 
Patent log, 486 
Path of a jet, 54 
Peak load, 382 
Pelton wheel, 441, 442 
Pendulum, hydrometric, 324 
Penstock, 383, 385, 392 
Perimeter, wetted, 272 
Physical properties of water, 3 
Piers, 342 

Piezometer, 230, 234, 238, 246 
Pipes, 42, 143, 211-271, 530 
curves in, 219, 410 
friction factors for, 217, 269 
friction heads for, 218, 270 
smooth, 67 
Piston pump, 512 
Pitometer, 93, 247 
Pitot’s tube, 101, 247, 324, 486 
Plates, moving, 408, 488 
Plunger pumps, 513 
Pneumatic turbine, 476 
Poiseuille’s law, 268 
Poncelet wheel, 439 
Potential energy, 3, 45 
Power, 3, 56, 452, 506 
dynamometer, 387 
Press, hydrostatic, 24 
Pressure, atmospheric, 7, 8, 20, 41 
center of, 34, 36 
dynamic, 399-431 
energy of, 177 
flow under, 49 
gages, 81, 85 
horizontal, 32 
measurement of, 81-88, 82 


Pressure, negative, 69 
normal, 31 
of waves, 409, 502 
on dams, 39, 40 
regulator, 247 
submerged body, 31 
transmission of, 23 
unit of, 2, 20 
Pressure gage, 8, 81, 85 
head, 25, 26, 41, 68, 244 
regulator, 247, 249 
Price current meter, 97 
Probable errors, 130 
Prony brake, 389 
Propeller, 492, 496 
Propulsion, work in, 490 
Pulsometer, 529 
Pumps, 7, 377, 504 
Pumping through hose, 534 
Pumping through pipes, 530 
Pumping engines, 517 
Poppet valve, 515 

Radius, hydraulic, 272 
gyration, 499 
Ram, hydraulic, 524, 526 
in pipes, 412 
Range of a jet, 54, 199 
Rain gage, 365 
Rainfall, 365 
Rating curve, 330 
Rating a meter, 100 
Reaction, 58, 400 
experiments on, 403 
turbines, 457-467, 521 
wheel, 430, 453 
Reciprocating pumps, 527 
Recording apparatus, 77, 91 
Rectangular conduits, 282, 284 
orifices, 122, 127, 139 
Reducer, 240 
Regulating devices, 248 
Regulator, pressure, 247 
Relative capacities of pipes, 235 
velocity, 60, 425 
Relief valves, 249 
Reservoirs, 78, 380 



Index 


563 


Resistance of plates, 487 
of ships, 486 
Reversibility, 528 
Revolving tubes, 429 
vanes, 423 
vessel, 62 

Rife hydraulic engine, 526 
Ring nozzle, 198 
Rivers, 318-364 
River water, 4, 7, 17 
Riveted pipes, 260, 296 
Rochester water pipe, 242 
Rod float, 323 
Rolling of a ship, 31, 498 
Roman aqueducts, 13, 265 
pipes, 13, 211 
Rotary pumps, 527 
Rounded crests, 160 
orifices, 109, 128 
Rudder, action of, 500 
Runoff, 372 


Salt water, 7, 19 
Sand, weight in water, 28 
filter bed, 250 
Screens, 308, 310 
Screw propeller, 495 
turbine, 477 
Seepage, 376 
Sewage, 7, 530 
Sewers, 289, 318 
Ships, 485-503 
Shock, 434 
Short pipes, 230 
tube, 184 
Siamese joint, 534 
Siphon, 239, 260 
Skin of water, 4, 79 
Slip of a ship, 495, 496 
Slope, 273, 317 
Small pipes, 268 
Smooth nozzle, 198 
pipes, 67 
Snow, 372 

Sound, velocity of, 28 
Specific gravity, 42 
Speed of wheels, 428, 437 


Speed of ships, 486 
of turbines, 457, 461 
Sphere, 29, 33 

Square vertical orifices, 120, 139 
Squares, table of, 545, 548 
Stability of dams, 40 
of flotation, 29, 497 
Standard orifice, 186 
tube, 184 
weirs, 141 
Standpipe, 213 
Statical moment, 37 
Steady flow, 273, 318, 539 
Steamer, coal used by, 491 
Steam plants, 381 
Steel pipes, 295, 296 
Stone, weight of, 28 
Storage of water, 378, 381 
Strength of pipes, 34, 42 
Submerged bodies, 31 
dams, 342 
orifices, 109, 126 
surfaces, 487 
tubes, 194 
turbines, 458 
weirs, 157 

Sub-surface float, 322, 333, 336 
velocities, 323, 330 
Suction, 8, 504, 506 
Suction pump, 504, 507 
Sudbury conduit, 301, 314 
Suppressed weirs, 152, 175 
Suppression of contraction, 127 
Surface curve, 167, 348 
float, 322 
velocity, 321, 330 
Surfaces, center of pressure, 36, 39 
jets upon, 58, 405 
pressure on, 32, 399 
Syringe, 505 


Tables, x, 545~556 
Tank, 76, 125, 384 
Temperature, 6, 130, 547 
Test of motors, 388 
pumping engines, 519 
turbines, 392, 481 





Index 


564 

Theoretical hydraulics, 44-74 
Theoretic discharge, 65 
velocity, 46, 52 
Thermal heat unit, 518 
Throttle valve, 223 
Tidal bore, 350 
waves, 397, 501 
Tide gate, 38 
Tides, 397, 452, 501 
Time, 2, 18 

Transmission of pressures, 24 
Transporting capacity, 294, 339 
Trapezoidal conduits, 286 
weirs, 170 

Triangular orifices, no 
Triangular weirs, 168 
Trigonometric functions, 545, 552 
Triple nozzle, 444 
Troughs, 272 
Tubes, 101, 177-210, 429 
Tubercules in pipes, 259, 262 
Tunnel, Niagara, 478 
Turbines, 14, 383, 453-484, 528 
Tutton’s formula, 304 
Twin screws, 496 
turbines, 461 

Undershot wheels, 439, 450 
Uniform flow, 67, 204, 274 
Unit of heat, 518 
Units of measure, 1, 18, 547 
Unsteady flow, 334 
Uplift, dams, 40 

Vacuum, 7, 13, 188, 504 
compound tube, 188 
pumps, 517 
standard tube, 187 
turbines, 475 
Valves, 223, 248, 251 
Vanes, 417, 440, 469 
in motion, 423 
revolving, ^29 

Variations in discharge, 130, 337 
in rainfall, 368 
Velocities in a cross-section, 204, 310, 320 


Velocity, 2, 18, 44 
absolute, 60 
coefficient of, 113 
critical, 269 
curves of, 204 
from orifices, 47 
in conduits, 275 
in pipes, 204, 267, 274 
- in rivers, 321 
mean, 274, 275 

measurement of, 95, 96, 101, 322 
of approach, 51, 145-153 
of sound and stress, 10, 21 
of the bore, 352 , 

of waves, 501 
relative, 60 

to move materials, 301, 339 
Velocity-head, 47, 68 
Venturi water meter, 89, 205 
Vermeule’s formula, 371 
Vertical jets, 46, 114, 199, 219 
orifices, 116, 118, 121 
Vertical turbines, 451 
wheels, 444 

Vessel, emptying of, 69 
moving, 61 
revolving, 63 
Viscous flow, 541 
Vortex whirl, 71 

Waste of water, 246 
weirs, 162 

Water, barometer, 8, 20, 507 
boiling point of, 8 
compressibility, 9 
distilled, 6, 19 
dynamic pressure, 58, 399 
freezing of, 4, 5, 18 
hammer, 248, 412 
mains, 227, 251 
maximum density, 4, 6 
measurement of, 77, 132, 384 
meters, 88 

physical properties, 3-20 
pipes, 34, 42, 211-271 
power, 381-398 
pressure of, 2, 18, 23 





Index 


565 


Water, storm, 373 
supply, 365-381 
surface of, 4, 24 
vapor, 507 
waste of, 251 
weight of, 6, 19 
Water-pressure engine, 451 
Watershed, 370 
Water wheels, 423, 432-452 
Waterwitch, 493 
Waves, 351, 408, 501 
Weighing water, 77, 385 
Weight of ice, 7, 19 
masonry, 40 
mercury, 8, 83 
sand, 28 
sewage, 7 

submerged bodies, 27 
water, 6, 19, 485 
Weirs, 80, 141-176, 386 
Wetted perimeter, 272 
Wheel pit, 478 
Wheels, breast, 426, 450 
horizontal, 445, 459 
impulse, 443, 448 


Wheels, overshot, 435, 449 
reaction, 430, 453, 473 
turbine, 453-484 
undershot, 434, 450 
vertical, 443, 460 
Whirl at orifice, 71 
Wide crests, 161 

Williams and Hazen’s formula, 304 
Wind, 322, 328, 332, 370 
Wire, line, 541 
Wood conduits, 281, 297 
Wood pipes, 263, 295 
Work, defined, 3, 382 
friction, 216, 276 
motors, 433, 481 
propulsion, 490 
pumping, 505 
ships, 490, 494 
vanes, 421, 425 
units of, 3, 18, 547 


Yield of watershed, 378 
Young man, 17, 513, 544 











































































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